John  S.   Prell 


. 

THEORETICAL  ASTRONOMY 


RELATING    TO    THE 


MOTIONS  OF  THE  HEAVENLY  BODIES 


REVOLVING  AROUND  THE  SUN  IN  ACCORDANCE  WITH 
THE  LAW  OF  UNIVERSAL  GRAVITATION 


EMBRACING 

A  SYSTEMATIC  DERIVATION  OF  THE  FORMULA  FOR  THE  CALCULATION    OF    THE  GEOCENTRIC   AND  UELIO- 
CENTRIC   PLACES,  FOR  THE   DETERMINATION   OF  THE   ORBITS   OF  PLANETS    ANi»  COMETS,  FOB 
THE  CORRECTION   OF  APPROXIMATE   ELEMENTS,  AND  FOR  THE   COMPUTATION  OF 
SPECIAL  PERTURBATIONS;  TOGETHER  WITH  THE  THEORY  Of  THE  COMBI- 
NATION OF  OBSERVATIONS  AND  THE  METHOD  OF  LEAST  SQUARES. 


Utiih  Jtummral  femgte  and 

JOHN  S.  PRELL 

Civil  &  Mechanical  Engineer. 

SAN  FRANCISCO,  CAL. 

BY 

JAMES    C.   WATSON 

DIRECTOR  OF  THE  OBSERVATORY  AT  ANN  ARBOR,  AND  PROFESSOR  OF  ASTRONOMY  IN  THE 
UNIVERSITY  OF  MICHIGAN 


PHILADELPHIA: 

J.  B.  LIPPINCOTT   COMPANY. 

LONDON:  36  SOUTHAMPTON  STREET,  COVENT  GARDEN. 
1900. 


Entered,  according  to  Act  of  Congress,  in  the  year  1868  '-v 
J.  B.  LIPPINCOTT   &   CO., 

in  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Eastern  District 
of  Pennsylvania. 


Copyright,  1896,  by  ANNETTE  H.  WATSON. 


GIFT 


555 

JOHN  S.  PRELL 

Civil  &  Mechanical  Engineer. 

SAN  FRANCISCO,  CAL. 
PREFACE. 


THE  discovery  of  the  great  law  of  nature,  the  law  of  gravitation,  by 
NEWTON,  prepared  the  way  for  the  brilliant  achievements  which  have 
distinguished  the  history  of  astronomical  science.  A  first  essential,  how- 
ever, to  the  solution  of  those  recondite  problems  which  were  to  exhibit 
the  effect  of  the  mutual  attraction  of  the  bodies  of  our  system,  was  the 
development  of  the  infinitesimal  calculus;  and  the  labors  of  those  who 
devoted  themselves  to  pure  analysis  have  contributed  a  most  important 
part  in  the  attainment  of  the  high  degree  of  perfection  which  character- 
izes the  results  of  astronomical  investigations.  Of  the  earlier  efforts  to 
develop  the  great  results  following  from  the  law  of  gravitation,  those  of 
EULER  stand  pre-eminent,  and  the  memoirs  which  he  published  have, 
in  reality,  furnished  the  germ  of  all  subsequent  investigations  in 
celestial  mechanics.  In  this  connection  also  the  names  of  BERNOTJILLI, 
CLAIRATJT,  and  D'ALEMBERT  deserve  the  most  honorable  mention  as 
having  contributed  also,  in  a  high  degree,  to  give  direction  to  the  inves- 
tigations which  were  to  unfold  so  many  mysteries  of  nature.  By  means 
of  the  researches  thus  inaugurated,  the  great  problems  of  mechanics 
were  successfully  solved,  many  beautiful  theorems  relating  to  the  planet- 
ary motions  demonstrated,  and  many  useful  formulae  developed. 

It  is  true,  however,  that  in  the  early  stage  of  the  science  methods 
were  developed  which  have  since  been  found  to  be  impracticable,  even 
if  not  erroneous;  still,  enough  was  effected  to  direct  attention  in  the 
proper  channel,  and  to  prepare  the  way  for  the  more  complete  labors  of 
LAGE-O.NGE  and  LAPLACE.  The  genius  and  the  analytical  skill  of  these 
extraordinary  men  gave  to  the  progress  of  Theoretical  Astronomy  the 
most  rapid  strides ;  and  the  intricate  investigations  which  they  success- 
fully performed,  served  constantly  to  educe  new  discoveries,  so  that  of 
all  the  problems  relating  to  the  mutual  attraction  of  the  several  planets 

3 

M718385 


4  PREFACE. 

but  little  more  remained  to  be  accomplished  by  their  successors  than  to 
develop  and  simplify  the  methods  which  they  made  known,  and  to  intro- 
duce such  modifications  as  should  be  indicated  by  experience  or  rendered 
possible  by  the  latest  discoveries  in  the  domain  of  pure  analysis. 

The  problem  of  determining  the  elements  of  the  orbit  of  a  comet 
moving  in  a  parabola,  by  means  of  observed  places,  which  had  been 
considered  by  NEWTON,  EULER,  BOSCOVICH,  LAMBERT,  and  others, 
received  from  LAGRANGE  and  LAPLACE  the  most  careful  consideration 
in  the  light  of  all  that  had  been  previously  done.  The  solution  given 
by  the  former  is  analytically  complete,  but  far  from  being  practically 
complete ;  that  given  by  the  latter  is  especially  simple  and  practical  so 
far  as  regards  the  labor  of  computation ;  but  the  results  obtained  by  it 
are  so  affected  by  the  unavoidable  errors  of  observation  as  to  be  often 
little  more  than  rude  approximations.  The  method  which  was  found  to 
answer  best  in  actual  practice,  was  that  proposed  by  OLBERS  in  his 
work  entitled  Leichteste  und  bequemste  Methode  die  Bahn  eines  Cometen 
zu  berechnen,  in  which,  by  making  use  of  a  beautiful  theorem  of  para- 
bolic motion  demonstrated  by  EULER  and  also  by  LAMBERT,  and  by 
adopting  a  method  of  trial  and  error  in  the  numerical  solution  of 
certain  equations,  he  was  enabled  to  effect  a  solution  which  could  be 
performed  with  remarkable  ease.  The  accuracy  of  the  results  obtained 
by  OLBERS'S  method,  and  the  facility  of  its  application,  directed  the 
attention  of  LEGENDRE,  IVORY,  GAUSS,  and  ENCKE  to  this  subject,  and 
by  them  the  method  was  extended  and  generalized,  and  rendered  appli 
cable  in  the  exceptional  cases  in  which  the  other  methods  failed. 

It  should  be  observed,  however,  that  the  knowledge  of  one  element, 
the  eccentricity,  greatly  facilitated  the  solution ;  and,  although  elliptic 
elements  had  been  computed  for  some  of  the  comets,  the  first  hypothesis 
was  that  of  parabolic  motion,  so  that  the  subsequent  process  required 
simply  the  determination  of  the  corrections  tc  be  applied  to  these  ele- 
ments in  order  to  satisfy  the  observations.  The  more  difficult  problem 
of  determining  all  the  elements  of  planetary  motion  directly  from  three 
observed  places,  remained  unsolved  until  the  discovery  of  Ceres  by 
PIAZZI  in  1801,  by  which  the  attention  of  GAUSS  was  directed  to  this 
subject,  the  result  of  which  was  the  subsequent  publication  of  his 
Theoria  Motus  Corporum  Cwlestium,  a  most  able  work,  in  which  he  gave 
to  the  world,  in  a  finished  form,  the  results  of  many  years  of  attention 


PRPJFACE.  5 

to  the  subject  of  which  it  treats.  His  method  for  .determining  all  the 
elements  directly  from  given  observed  places,  as  given  in  the  Theoria 
Motus,  and  as  subsequently  given  in  a  revised  form  by  ENCKE,  leaves 
scarcely  any  thing  to  be  desired  on  this  topic.  In  the  same  work  he 
gave  the  first  explanation  of  the  method  of  least  squares,  a  method 
which  has  been  of  inestimable  service  in  investigations  depending  on 
observed  data. 

The  discovery  of  the  minor  planets  directed  attention  also  to  the 
methods  of  determining  their  perturbations,  since  those  applied  in  the 
case  of  the  major  planets  were  found  to  be  inapplicable.  For  a  long 
time  astronomers  were  content  simply  to  compute  the  special  perturba- 
tions of  these  bodies  from  epoch  to  epoch,  and  it  was  not  until  the  com- 
mencement of  the  brilliant  researches  by  HANSEN  that  serious  hopes 
were  entertained  of  being  able  to  compute  successfully  the  general  per- 
turbations of  these  bodies.  By  devising  an  entirely  new  mode  of  con- 
sidering the  perturbations,  namely,  by  determining  what  may  be  called 
the  perturbations  of  the  time,  and  thus  passing  from  the  undisturbed 
place  to  the  disturbed  place,  and  by  other  ingenious  analytical  and 
mechanical  devices,  he  succeeded  in  effecting  a  solution  of  this  most 
difficult  problem,  and  his  latest  works  contain  all  the  formulse  which  are 
required  for  the  cases  actually  occurring.  The  refined  and  difficult 
analysis  and  the  laborious  calculations  involved  were  such  that,  even 
after  HANSEN'S  methods  were  made  known,  astronomers  still  adhered  to 
the  method  of  special  perturbations  by  the  variation  of  constants  as 
developed  by  LAGRANGE. 

The  discovery  of  Astrcea  by  HENCKE  was  speedily  followed  by  the 
discovery  of  other  planets,  and  fortunately  indeed  it  so  happened  that 
the  subject  of  special  perturbations  was  to  receive  a  new  improvement. 
The  discovery  by  BOND  and  ENCKE  of  a  method  by  which  we  determine 
at  once  the  variations  of  the  rectangular  co-ordinates  of  the  disturbed 
body  by  integrating  the  fundamental  equations  of  motion  by  means  of 
mechanical  quadrature,  directed  the  attention  of  HANSEN  to  this  phase 
of  the  problem,  and  soon  after  he  gave  formulse  for  the  determination 
of  the  perturbations  of  the  latitude,  the  mean  anomaly,  and  the  loga- 
rithm of  the  radius-vector,  which  are  exceedingly  convenient  in  the 
process  of  integration,  and  which  have  been  found  to  give  the  most 
satisfactory  results.  The  formulse  for  the  perturbations  of  the  latitude, 


6  PREFACE. 

true  longitude,  and  radius-vector,  to  be  integrated  in  the  same  manner, 
were  afterwards  given  by  BRUNNOW. 

Having  thus  stated  briefly  a  few  historical  facts  relating  to  the 
problems  of  theoretical  astronomy,  I  proceed  to  a  statement  of  the 
object  of  this  work.  The  discovery  of  so  many  planets  and  comets  has 
furnished  a  wide  field  for  exercise  in  the  calculations  relating  to  their 
motions,  and  it  has  occurred  to  me  that  a  work  which  should  contain  a 
development  of  all  the  formulae  required  in  determining  the  orbits  of  the 
heavenly  bodies  directly  from  given  observed  places,  and  in  correcting 
these  orbits  by  means  of  more  extended  discussions  of  series  of  observa- 
tions, including  also  the  determination  of  the  perturbations,  together 
with  a  complete  collection  of  auxiliary  tables,  and  also  such  practical 
directions  as  might  guide  the  inexperienced  computer,  might  add  very 
materially  to  the  progress  of  the  science  by  attracting  the  attention  of  a 
greater  number  of  competent  computers.  Having  carefully  read  the 
works  of  the  great  masters,  my  plan  was  to  prepare  a  complete  work  on 
this  subject,  commencing  with  the  fundamental  principles  of  dynamics, 
and  systematically  treating,  from  one  point  of  view,  all  the  problems 
presented.  The  scope  and  the  arrangement  of  the  work  will  be  best 
understood  after  an  examination  of  its  contents ;  and  let  it  suffice  to  add 
that  I  have  endeavored  to  keep  constantly  in  view  the  wants  of  the 
computer,  providing  for  the  exceptional  cases  as  they  occur,  and  giving 
all  the  formulae  which  appeared  to  me  to  be  best  adapted  to  the  problems 
under  consideration.  I  have  not  thought  it  worth  while  to  trace  out  the 
geometrical  signification  of  many  of  the  auxiliary  quantities  introduced. 
Those  who  are  curious  in  such  matters  may  readily  derive  many  beau- 
tiful theorems  from  a  consideration  of  the  relations  of  some  of  these 
auxiliaries.  For  convenience,  the  formulae  are  numbered  consecutively 
through  each  chapter,  and  the  references  to  those  of  a  preceding  chapter 
are  defined  by  adding  a  subscript  figure  denoting  the  number  of  the 
chapter. 

Besides  having  read  the  works  of  those  who  have  given  special  atten 
tion  to  these  problems,  I  have  consulted  the  Astronomische  Nachrichten, 
the  Astronomical  Journal,  and  other  astronomical  periodicals,  in  which 
'is  to  be  found  much  valuable  information  resulting  from  the  experi- 
ence of  those  wrho  have  been  or  are  now  actively  engaged  in  astro- 
nomical pursuits.  I  must  also  express  my  obligations  to  the  publishers, 


PREFACE.  7 

Messrs.  J.  B.  LIPPINCOTI  &  Co.,  for  the  generous  interest  which  they 
have  manifested  in  the  publication  of  the  work,  and  also  to  Dr.  B.  A. 
GOULD,  of  Cambridge,  Mass.,  and  to  Dr.  OPPOLZER,  of  Vienna,  for 
valuable  suggestions. 

For  the  determination  of  the  time  from  the  perihelion  and  of  the  true 
anomaly  in  very  eccentric  orbits  I  have  given  the  method  proposed  by 
BESSEL  in  the  Monatliche  Correspondent,  vol.  xii., — the  tables  for  which 
were  subsequently  given  by  BRUNNOW  in  his  Astronomical  Notices, — and 
also  the  method  proposed  by  GAUSS,  but  in  a  more  convenient  form. 
For  obvious  reasons,  I  have  given  the  solution  for  the  special  case  of 
parabolic  motion  before  completing  the  solution  of  the  general  problem 
of  finding  all  of  the  elements  of  the  orbit  by  means  of  three  observed 
places.  The  differential  formulae  and  the  other  formulae  for  correcting 
approximate  elements  are  given  in  a  form  convenient  for  application, 
and  the  formulas  for  finding  the  chord  or  the  time  of  describing  the 
subtended  arc  of  the  orbit,  in  the  case  of  very  eccentric  orbits,  will  be 
found  very  convenient  in  practice. 

I  have  given  a  pretty  full  development  of  the  application  of  the 
theory  of  probabilities  to  the  combination  of  observations,  endeavoring 
to  direct  the  attention  of  the  reader,  as  far  as  possible,  to  the  sources  of 
error  to  be  apprehended  and  to  the  most  advantageous  method  of  treat- 
ing the  problem  so  as  to  eliminate  the  effects  of  these  errors.  For  the 
rejection  of  doubtful  observations,  according  to  theoretical  considerations, 
I  have  given  the  simple  formula,  suggested  by  CHAUVENET,  which  fol 
lows  directly  from  the  fundamental  equations  for  the  probability  of 
errors,  and  which  will  answer  for  the  purposes  here  required  as  well  as 
the  more  complete  criterion  proposed  by  PEIRCE.  In  the  chapter 
devoted  to  the  theory  of  special  perturbations  I  have  taken  particular 
pains  to  develop  the  whole  subject  in  a  complete  and  practical  form, 
keeping  constantly  in  view  the  requirements  for  accurate  and  convenient 
numerical  application.  The  time  is  adopted  as  the  independent  variable 
in  the  determination  of  the  perturbations  of  the  elements  directly,  since 
experience  has  established  the  convenience  of  this  form ;  and  should  it 
be  desired  to  change  the  independent  variable  and  to  use  the  differential 
coefficients  with  respect  to  the  eccentric  anomaly,  the  equations  between 
this  function  and  the  mean  motion  will  enable  us  to  effect  readily  the 
required  transformation. 


8  PREFACE. 

The  numerical  examples  involve  data  derived  from  actual  observa- 
tions, and  care  has  been  taken  to  make  them  complete  in  every  respect, 
so  as  to  serve  as  a  guide  to  the  efforts  of  those  not  familiar  with  these 
calculations ;  and  when  different  fundamental  planes  are  spoken  of,  it  is 
presumed  that  the  reader  is  familiar  with  the  elements  of  spherical 
astronomy,  so  that  it  is  unnecessary  to  state,  in  all  cases,  whether  the 
centre  of  the  sphere  is  taken  at  the  centre  of  the  earth,  or  at  any  other 
point  in  space. 

The  preparation  of  the  Tables  has  cost  me  a  great  amount  of  labor, 
logarithms  of  ten  decimals  being  employed  in  order  to  be  sure  of  the 
last  decimal  given.  Several  of  those  in  previous  use  have  been  recom- 
puted and  extended,  and  others  here  given  for  the  first  time  have  been 
prepared  with  special  care.  The  adopted  value  of  the  constant  of  the 
solar  attraction  is  that  given  by  GAUSS,  which,  as  will  appear,  is  not 
accurately  in  accordance  with  the  adoption  of  the  mean  distance  of  the 
earth  from  the  sun  as  the  unit  of  space;  but  until  the  absolute  value  of 
the  earth's  mean  motion  is  known,  it  is  best,  for  the  sake  of  uniformity 
and  accuracy,  to  retain  GAUSS'S  constant. 

The  preparation  of  this  work  has  been  effected  amid  many  intern;  p- 
tions,  and  with  other  labors  constantly  pressing  me,  by  which  the  progress 
of  its  publication  has  been  somewhat  delayed,  even  since  the  stereo- 
typing was  commenced,  so  that  in  some  cases  I  have  been  anticipated 
in  the  publication  of  formulae  which  would  have  here  appeared  for  the 
first  time.  I  have,  however,  endeavored  to  perform  conscientiously  the 
self-imposed  task,  seeking  always  to  secure  a  logical  sequence  in  the  de- 
velopment of  the  formulae,  to  preserve  uniformity  and  elegance  in  the 
notation,  and  to  elucidate  the  successive  steps  in  the  analysis,  so  that  the 
work  may  be  read  by  those  who,  possessing  a  respectable  mathematical 
education,  desire  to  be  informed  of  the  means  by  which  astronomers  are 
enabled  to  arrive  at  so  many  grand  results  connected  with  the  motions 
of  the  heavenly  bodies,  and  by  which  the  grandeur  and  sublimity  of 
creation  are  unveiled.  The  labor  of  the  preparation  of  the  work  will 
have  been  fully  repaid  if  it  shall  be  the  means  of  directing  a  more 
general  attention  to  the  study  of  the  wonderful  mechanism  of  the  hea- 
vens, the  contemplation  of  which  must  ever  serve  to  impress  upon  the 
mind  the  reality  of  the  perfection  of  the  OMNIPOTENT,  the  LIVING  GOD ! 

OBSEBVATOKY,  ANN  ARBOR,  June,  1867. 


CONTENTS. 


THEORETICAL  ASTRONOMY. 


CHAPTER  I. 

INVESTIGATION  OP  THE  FUNDAMENTAL  EQUATIONS  OF  MOTION,  AND  OF  THE  FOR- 
MULA FOR  DETERMINING,  FROM  KNOWN  ELEMENTS,  THE  HELIOCENTRIC  AND 
GEOCENTRIC  PLACES  OF  A  HEAVENLY  BODY,  ADAPTED  TO  NUMERICAL  COM- 
PUTATION FOR  CASES  OF  ANY  ECCENTRICITY  WHATEVER. 

PAOB 

Fundamental  Principles 15 

Attraction  of  Spheres 19 

Motions  of  a  System  of  Bodies '. 23 

Invariable  Plane  of  the  System 29 

Motion  of  a  Solid  Body 31 

The  Units  of  Space,  Time,  and  Mass 36 

Motion  of  a  Body  relative  to  the  Sun 38 

Equations  for  Undisturbed  Motion 42 

Determination  of  the  Attractive  Force  of  the  Sun 49 

Determination  of  the  Place  in  an  Elliptic  Orbit 53 

Determination  of  the  Place  in  a  Parabolic  Orbit 59 

Determination  of  the  Place  in  a  Hyperbolic  Orbit 65 

Methods  for  finding  the  True  Anomaly  and  the  Time  from  the  Perihelion  in  the 

case  of  Orbits  of  Great  Eccentricity 70 

Determination  of  the  Position  in  Space 81 

Heliocentric  Longitude  and  Latitude 83 

Reduction  to  the  Ecliptic 85 

Geocentric  Longitude  and  Latitude 86 

Transformation  of  Spherical  Co-ordinates 87 

Direct  Determination  of  the  Geocentric  Eight  Ascension  and  Declination 90 

.Reduction  of  the  Elements  from  one  Epoch  to  another 99 

Numerical  Examples 103 

Interpolation < 112 

Time  of  Opposition 114 


10  CONTENTS. 


CHAPTER  II. 

INVESTIGATION  OF  THE  DIFFERENTIAL  FORMULAE  WHICH  EXPRESS  THE  RELATION 
BETWEEN  THE  GEOCENTRIC  OR  HELIOCENTRIC  PLACES  OF  A  HEAVENLY  BODY 
AND  THE  VARIATIONS  OF  THE  ELEMENTS  OF  ITS  ORBIT. 

PAGB 

Variation  of  the  Eight  Ascension  and  Declination 118 

Case  of  Parabolic  Motion  125 

Case  of  Hyperbolic  Motion 128 

Case  of  Orbits  differing  but  little  from  the  Parabola 130 

Numerical  Examples 135 

Variation  of  the  Longitude  and  Latitude 143 

The  Elements  referred  to  the  same  Fundamental  Plane  as  the  Geocentric  Places  149 

Numerical  Example 150 

Plane  of  the  Orbit  taken  as  the  Fundamental  Plane  to  which  the  Geocentric 

Places  are  referred 153 

Numerical  Example 159 

Variation  of  the  Auxiliaries  for  the  Equator 163 


CHAPTER  III. 

INVESTIGATION  OF  FORMULA  FOR  COMPUTING  THE  ORBIT  OF  A  COMET  MOVING 
IN  A  PARABOLA,  AND  FOR  CORRECTING  APPROXIMATE  ELEMENTS  BY  THE 
VARIATION  OF  THE  GEOCENTRIC  DISTANCE. 

Correction  of  the  Observations  for  Parallax 167 

Fundamental  Equations 169 

Particular  Cases 172 

Ratio  of  Two  Curtate  Distances 178 

Determination  of  the  Curtate  Distances 181 

Relation  between  Two  Radii- Vectores,  the  Chord  joining  their  Extremities,  and 

the  Time  of  describing  the  Parabolic  Arc 184 

Determination  of  the  Node  and  Inclination 192 

Perihelion  Distance  and  Longitude  of  the  Perihelion 194 

Time  of  Perihelion  Passage 195 

Numerical  Example 199 

Correction  of  Approximate  Elements  by  varying  the  Geocentric  Distance 208 

Numerical  Example 213 


CHAPTER  IV. 

DETERMINATION,   FROM  THREE    COMPLETE    OBSERVATIONS,    OF    THE    ELEMENTS    OF 
THE   ORBIT   OF   A  HEAVENLY  BODY,   INCLUDING   THE    ECCENTRICITY   OR    FORM 


OF  THE  CONIC  SECTION. 


Reduction  of  the  Data 220 

Corrections  for  Parallax 223 


CONTENTS.  ll 

PAOK 

Fundamental  Equations 225 

Formulae  for  the  Curtate  Distances 228 

Modification  of  the  Formulae  in  Particular  Cases 231 

Determination  of  the  Curtate  Distance  for  the  Middle  Observation 236 

Case  of  a  Double  Solution 239 

Posjtion  indicated  by  the  Curvature  of  the  Observed  Path  of  the  Body 242 

Formulae  for  a  Second  Approximation 243 

Formulas  for  finding  the  Katio  of  the  Sector  to  the  Triangle 247 

Final  Correction  for  Aberration 257 

Determination  of  the  Elements  of  the  Orbit 259 

Numerical  Example 264 

Correction  of  the  First  Hypothesis 278 

Approximate  Method  of  finding  the  Katio  of  the  Sector  to  the  Triangle 279 


CHAPTER  V. 

DETERMINATION    OF    THE    ORBIT    OP    A    HEAVENLY    BODY    FROM    FOUR    OBSERVA- 
TIONS,  OF  WHICH  THE  SECOND  AND  THIRD  MUST    BE  COMPLETE. 

Fundamental  Equations 282 

Determination  of  the  Curtate  Distances , 289 

Successive  Approximations 293 

Determination  of  the  Elements  of  the  Orbit 294 

Numerical  Example 294 

Method  for  the  Final  Approximation 307 


CHAPTER  VI. 

INVESTIGATION   OF  VARIOUS  FORMULAE    FOR    THE    CORRECTION    OF    THE  APPROXI- 
MATE  ELEMENTS  OF  THE  ORBIT  OF  A  HEAVENLY  BODY. 

Determination  of  the  Elements  of  a  Circular  Orbit 311 

Variation  of  Two  Geocentric  Distances 313 

Differential  Formulae 318 

Plane  of  the  Orbit  taken  as  the  Fundamental  Plane 320 

Variation  of  the  Node  and  Inclination 324 

Variation  of  One  Geocentric  Distance 328 

Determination  of  the  Elements  of  the  Orbit  by  means  of  the  Co-ordinates  and 

Velocities 332 

Correction  of  the  Ephemeris 335 

Final  Correction  of  the  Elements 338 

Kelation  between  Two  Places  in  the  Orbit 339 

Modification  when  the  Semi-Transverse  Axis  is  very  large 341 

Modification  for  Hyperbolic  Motion 346 

Variation  of  the  Semi-Transverse  Axis  and  Eatio  of  Two  Curtate  Distances 349 


12  CONTENTS. 


PAQB 


Variation  of  the  Geocentric  Distance  and  of  the  Reciprocal  of  the  Semi-Trans- 
verse Axis 352 

Equations  of  Condition 353 

Orbit  of  a  Comet 355 

Variation  of  Two  Eadii-Vectores 357 


CHAPTER  VII. 

METHOD  OF  LEAST  SQUARES,  THEORY  OP  THE  COMBINATION  OF  OBSERVATIONS, 
AND  DETERMINATION  OF  THE  MOST  PROBABLE  SYSTEM  OF  ELEMENTS  FROM 
A  SERIES  OF  OBSERVATIONS. 

Statement  of  the  Problem 360 

Fundamental  Equations  for  the  Probability  of  Errors 362 

Determination  of  the  Form  of  the  Function  which  expresses  the  Probability  ...  363 

The  Measure  of  Precision,  and  the  Probable  Error 366 

Distribution  of  the  Errors 367 

The  Mean  Error,  and  the  Mean  of  the  Errors 368 

The  Probable  Error  of  the  Arithmetical  Mean 370 

Determination  of  the  Mean  and  Probable  Errors  of  Observations 371 

Weights  of  Observed  Values 372 

Equations  of  Condition 376 

Normal  Equations 378 

Method  of  Elimination 380 

Determination  of  the  Weights  of  the  Resulting  Values  of  the  Unknown  Quanti- 
ties   386 

Separate  Determination  of  the  Unknown  Quantities  and  of  their  Weights 392 

Eelation  between  the  Weights  and  the  Determinants 396 

Case  in  which  the  Problem  is  nearly  Indeterminate 398 

Mean  and  Probable  Errors  of  the  Results 399 

Combination  of  Observations 401 

Errors  peculiar  to  certain  Observations 408 

Rejection  of  Doubtful  Observations 410 

Correction  of  the  Elements 412 

Arrangement  of  the  Numerical  Operations 415 

Numerical  Example 418 

Case  of  very  Eccentric  Orbits 423 


CHAPTER  VIII. 

INVESTIGATION  OF  VARIOUS  FORMULA  FOR   THE  DETERMINATION  OF  THE  SPECIAL 
PERTURBATIONS  OF  A  HEAVENLY  BODY. 

Fundamental  Equations 426 

Statement  of  the  Problem '. 428 

Variation  of  Co-ordinates 429 


CONTENTS.  13 

PAQB 

Mechanical  Quadrature 433 

The  Interval  for  Quadrature 443 

Mode  of  effecting  the  Integration 445 

Perturbations  depending  on  the  Squares  and  Higher  Powers  of  the  Masses 446 

Numerical  Example 448 

Change  of  the  Equinox  and  Ecliptic 455 

Determination  of  New  Osculating  Elements 459 

Variation  of  Polar  Co-ordinates 462 

Determination  of  the  Components  of  the  Disturbing  Force 467 

Determination  of  the  Heliocentric  or  Geocentric  Place 471 

Numerical  Example 474 

Change  of  the  Osculating  Elements 477 

Variation  of  the  Mean  Anomaly,  the  Radius- Vector,  and  the  Co-ordinate  2......  480 

Fundamental  Equations 483 

Determination  of  the  Components  of  the  Disturbing  Force 489 

Case  of  very  Eccentric  Orbits 493 

Determination  of  the  Place  of  the  Disturbed  Body 495 

Variation  of  the  Node  and  Inclination 502 

Numerical  Example 505 

Change  of  the  Osculating  Elements 510 

Variation  of  Constants 516 

Case  of  very  Eccentric  Orbits 523 

Variation  of  the  Periodic  Time 526 

Numerical  Example 529 

Formulae  to  be  used  when  the  Eccentricity  or  the  Inclination  is  small 533 

Correction  of  the  Assumed  Value  of  the  Disturbing  Mass 535 

Perturbations  of  Comets 536 

Motion  about  the  Common  Centre  of  gravity  of  the  Sun  and  Planet 537 

Keduction  of  the  Elements  to  the  Common  Centre  of  Gravity  of  the  Sun  and 

Planet  538 

Keduction  by  means  of  Differential  Formulae 540 

Near  Approach  of  a  Comet  to  a  Planet 546 

The  Sun  may  be  regarded  as  the  Disturbing  Body 548 

Determination  of  the  Elements  of  the  Orbit  about  the  Planet 550 

Subsequent  Motion  of  the  Comet 551 

Effect  of  a  Resisting  Medium  in  Space 552 

Variation  of  the  Elements  on  account  of  the  Resisting  Medium 554 

Method  to  be  applied  when  no  Assumption  is  made  in  regard  to  the  Density  o* 

the  Ether....  5*6 


14  CONTENTS. 


TABLES. 


PAOl 

I.  Angle  of  the  Vertical  and  Logarithm  of  the  Earth's  Radius ~  561 

H.  For  converting  Intervals  of  Mean  Solar  Time  into  Equivalent  Intervals 

of  Sidereal  Time 563 

III.  For  converting  Intervals  of  Sidereal  Time  into  Equivalent  Intervals 

of  Mean  Solar  Time 564 

IV.  For  converting  Hours,  Minutes,  and  Seconds  into  Decimals  of  a  Day...  565 
V.  For  finding  the  Number  of  Days  from  the  Beginning  of  the  Year 565 

VI.  For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a 

Parabolic  Orbit 566 

VII.  For  finding  the  True  Anomaly  in  a  Parabolic  Orbit  when  v  is  nearly  180°  611 

VIII.  For  finding  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit 612 

IX.  For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  Orbits 

of  Great  Eccentricity 614 

X.  For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  El- 
liptic and  Hyperbolic  Orbits 618 

XI.  For  the  Motion  in  a  Parabolic  Orbit 619 

XII.  For  the  Limits  of  the  Boots  of  the  Equation  sin  (z1  —£)=m0  sin4  y! ...  622 

XIII.  For  finding  the  Eatio  of  the  Sector  to  the  Triangle 624 

XIV.  For  finding  the  Eatio  of  the  Sector  to  the  Triangle 629 

XV.  For  Elliptic  Orbits  of  Great  Eccentricity 632 

XVI.  For  Hyperbolic  Orbits 632 

XVII.  For  Special  Perturbations 633 

XVIII.  Elements  of  the  Orbits  of  the  Comets  which  have  been  observed 638 

XIX.  Elements  of  the  Orbits  of  the  Minor  Planets 646 

XX.  Elements  of  the  Orbits  of  the  Major  Planets 648 

XXI.  Constants,  &c 649 

EXPLANATION  OP  THE  TABLES 651 

APPENDIX. — Precession 657 

Nutation 658 

Aberration  659 

Intensity  of  Light 660 

Numerical  Calculations ..  662 


THEORETICAL  ASTRONOMY. 


CHAPTER  I. 

INVESTIGATION  OF  THE  FUNDAMENTAL  EQUATIONS  OF  MOTION,  AND  OF  THE  FOR- 
MULA FOB  DETERMINING,  FROM  KNOWN  ELEMENTS,  THE  HELIOCENTRIC  AND 
GEOCENTRIC  PLACES  OF  A  HEAVENLY  BODY,  ADAPTED  TO  NUMERICAL  COMPUTA- 
TION FOR  CASES  OF  ANY  ECCENTRICITY  WHATEVER. 

1.  THE  study  of  the  motions  of  the  heavenly  bodies  does  not  re- 
quire that  we  should  know  the  ultimate  limit  of  divisibility  of  the 
matter  of  which  they  are  composed, — whether  it  may  be  subdivided 
indefinitely,  or  whether  the  limit  is  an  indivisible,  impenetrable  atom. 
Nor  are  we  concerned  with  the  relations  which  exist  between  the 
separate  atoms  or  molecules,  except  so  far  as  they  form,  in  the  aggre- 
gate, a  definite  body  whose  relation  to  other  bodies  of  the  system  it 
is  required  to  investigate.  On  the  contrary,  in  considering  the  ope- 
ration of  the  laws  in  obedience  to  which  matter  is  aggregated  into 
single  bodies  and  systems  of  bodies,  it  is  sufficient  to  conceive  simply 
of  its  divisibility  to  a  limit  which  may  be  regarded  as  infinitesimal 
compared  with  the  finite  volume  of  the  body,  and  to  regard  the  mag- 
nitude of  the  element  of  matter  thus  arrived  at  as  a  mathematical 
point. 

An  element  of  matter,  or  a  material  body,  cannot  give  itself 
motion;  neither  can  it  alter,  in  any  manner  whatever,  any  motion 
which  may  have  been  communicated  to  it.  This  tendency  of  matter 
to  resist  all  changes  of  its  existing  state  of  rest  or  motion  is  known 
as  inertia,  and  is  the  fundamental  law  of  the  motion  of  bodies.  Ex- 
perience invariably  confirms  it  as  a  law  of  nature;  the  continuance  of 
motion  as  resistances  are  removed,  as  well  as  the  sensibly  unchanged 
motion  of  the  heavenly  bodies  during  many  centuries,  affording  the 

16 


16  THEORETICAL   ASTRONOMY. 

most  convincing  proof  of  its  universality.  Whenever,  therefore,  a 
material  point  experiences  any  change  of  its  state  as  respects  rest  or 
motion,  the  cause  must  be  attributed  to  the  operation  of  something 
external  to  the  element  itself,  and  which  we  designate  by  the  word 
force.  The  nature  of  forces  is  generally  unknown,  and  we  estimate 
them  by  the  effects  which  they  produce.  They  are  thus  rendered  com- 
parable with  some  unit,  and  may  be  expressed  by  abstract  numbers. 

2.  If  a  material  point,  free  to  move,  receives  an  impulse  by  virtue 
of  the  action  of  any  force,  or  if,  at  any  instant,  the  force  by  which 
motion  is  communicated  shall  cease  to  act,  the  subsequent  motion  of 
the  point,  according  to  the  law  of  inertia,  must  be  rectilinear  and 
uniform,  equal  spaces  being  described  in  equal  times.  Thus,  if  s,  v, 
and  t  represent,  respectively,  the  space,  the  velocity,  and  the  time,  the 
measure  of  v  being  the  space  described  in  a  unit  of  time,  we  shall 
have,  in  this  case, 

s  =  vt. 

It  is  evident,  however,  that  the  space  described  in  a  unit  of  time  will 
vary  with  the  intensity  of  the  force  to  which  the  motion  is  due,  and, 
the  nature  of  the  force  being  unknown,  we  must  necessarily  compare 
the  velocities  communicated  to  the  point  by  different  forces,  in  order 
to  arrive  at  the  relation  of  their  effects.  We  are  thus  led  to  regard 
the  force  as  proportional  to  the  velocity;  and  this  also  has  received 
the  most  indubitable  proof  as  being  a  law  of  nature.  Hence,  the 
principles  of  the  composition  and  resolution  of  forces  may  be  applied 
also  to  the  composition  and  resolution  of  velocities. 

If  the  force  acts  incessantly,  the  velocity  will  be  accelerated,  and 
the  force  which  produces  this  motion  is  called  an  accelerating  force. 
In  regard  to  the  mode  of  operation  of  the  force,  however,  we  may 
consider  it  as  acting  absolutely  without  cessation,  or  we  may  regard 
it  as  acting  instantaneously  at  successive  infinitesimal  intervals  repre- 
sented by  dt,  and  hence  the  motion  as  uniform  during  each  of  these 
intervals.  The  latter  supposition  is  that  which  is  best  adapted  to 
the  requirements  of  the  infinitesimal  calculus;  and,  according  to  the 
fundamental  principles  of  this  calculus,  the  finite  result  will  be  the 
same  as  in  the  case  of  a  force  whose  action  is  absolutely  incessant. 
Therefore,  if  we  represent  the  element  of  space  by  ds,  and  the  ele- 
ment of  time  by  dt,  the  instantaneous  velocity  will  be 

ds 
•-    V  =  ~dt> 

which  will  vary  from  one  instant  to  another. 


FUNDAMENTAL   PRINCIPLES.  17 

3.  Since  the  force  is  proportional  to  the  velocity,  its  measure  at 
any  instant  will  be  determined  by  the  corresponding  velocity.  If 
the  accelerating  force  is  constant,  the  motion  will  be  uniformly  accele- 
rated; and  if  we  designate  the  acceleration  due  to  the  force  by/,  the 
unit  of/  being  the  velocity  generated  in  a  unit  of  time,  we  shall  have 


If,  however,  the  force  be  variable,  we  shall  have,  at  any  instant, 

the  relation 

efc 
/~~  dt' 

the  force  being  regarded  as  constant  in  its  action  during  the  element 
of  time  dt.     The  instantaneous  value  of  v  gives,  by  differentiation, 

dv  _  d*s 
~dt~~W 
and  hence  we  derive 


so  that,  in  varied  motion,  the  acceleration  due  to  the  force  is  mea- 
sured by  the  second  differential  of  the  space  divided  by  the  square 
of  the  element  of  time. 

4.  By  the  mass  of  the  body  we  mean  its  absolute  quantity  of  mat- 
ter. The  density  is  the  mass  of  a  unit  of  volume,  and  hence  the 
entire  mass  is  equal  to  the  volume  multiplied  by  the  density.  If  it 
is  required  to  compare  the  forces  which  act  upon  different  bodies,  it 
is  evident  that  the  masses  must  be  considered.  If  equal  masses 
receive  impulses  by  the  action  of  instantaneous  forces,  the  forces 
acting  on  each  will  be  to  each  other  as  the  velocities  imparted;  and 
if  we  consider  as  the  unit  of  force  that  which  gives  to  a  unit  of  mass 
the  unit  of  velocity,  we  have  for  the  measure  of  a  force  F,  denoting 
the  mass  by  Jf, 

F  =  Mv. 

This  is  called  the  quantity  of  motion  of  the  body,  and  expresses  its 
capacity  to  overcome  inertia.  By  virtue  of  the  inert  state  of  matter, 
there  can  be  no  action  of  a  force  without  an  equal  and  contrary  re- 
action ;  for,  if  the  body  to  which  the  force  is  applied  is  fixed,  the 
equilibrium  between  the  resistance  and  the  force  necessarily  implies 
the  development  of  an  equal  and  contrary  force  ;  and,  if  the  body  be 
free  to  move,  in  the  change  of  state,  its  inertia  will  oppose  equal  and 


18  THEORETICAL   ASTRONOMY. 

contrary  resistance.     Hence,  as  a  necessary  consequence  of  inertia,  it 

follows  that  action  and  reaction  are  simultaneous,  equal,  and  contrary. 

If  the  body  is  acted  upon  by  a  force  such  that  the  motion  is  varied, 

the  accelerating  force  upon  each  element  of  its  mass  is  represented  by 

"7- ,  and  the  entire  motive  force  F  is  expressed  by 


M  being  the  sum  of  all  the  elements,  or  the  mass  of  the  body.   Since 

_ds_ 
this  gives 


which  ^is  the  expression  for  the  intensity  of  the  motive  force,  or  of 
the  force  of  inertia  developed.  For  the  unit  of  mass,  the  measure 
of  the  force  is 

d?s 


and  this,  therefore,  expresses  that  part  of  the  intensity  of  the  motive 
force  which  is  impressed  upon  the  unit  of  mass,  and  is  what  is  usually 
called  the  accelerating  force. 

5.  The  force  in  obedience  to  which  the  heavenly  bodies  perform 
their  journey  through  space,  is  known  as  the  attraction  of  gravitation; 
and  the  law  of  the  operation  of  this  force,  in  itself  simple  and  unique, 
has  been  confirmed  and  generalized  by  the  accumulated  researches  of 
modern  science.  Not  only  do  we  find  that  it  controls  the  motions  of 
the  bodies  of  our  own  solar  system,  but  that  the  revolutions  of  binary 
systems  of  stars  in  the  remotest  regions  of  space  proclaim  the  uni- 
versality of  its  operation.  It  unfailingly  explains  all  the  phenomena 
observed,  and,  outstripping  observation,  it  has  furnished  the  means 
of  predicting  many  phenomena  subsequently  observed.  The  law  of 
this  force  is  that  every  particle  of  matter  is  attracted  by  every  other 
particle  by  a  force  which  varies  directly  as  the  mass  and  inversely  as 
the  square  of  the  distance  of  the  attracting  particle. 

This  reciprocal  action  is  instantaneous,  and  is  not  modified,  in  any 
degree,  by  the  interposition  of  other  particles  or  bodies  of  matter.  It 
is  also  absolutely  independent  of  the  nature  of  the  molecules  them- 
selves, and  of  their  aggregation. 


ATTRACTION   OF   SPHERES.  19 

If  we  consider  two  bodies  the  masses  of  which  are  m  and  m',  and 
whose  magnitudes  are  so  small,  relatively  to  their  mutual  distance  />, 
that  we  may  regard  them  as  material  points,  according  to  the  law  of 
gravitation,  the  action  of  m  on  each  molecule  or  unit  of  m'  will  be 

—  ,  and  the  total  force  on  m!  will  be 

m'™. 
/>' 

The  action  of  mf  on  each  molecule  of  m  will  be  expressed  by  —  ,  and 

its  total  action  by 

m' 

™-r- 

The  absolute  or  moving  force  with  which  the  masses  m  and  mf  tend 
toward  each  other  is,  therefore,  the  same  on  each  body,  which  result 
is  a  necessary  consequence  of  the  equality  of  action  and  reaction. 
The  velocities,  however,  with  which  these  bodies  would  approach 
each  other  must  be  different,  the  velocity  of  the  smaller  mass  exceed- 
ing that  of  the  greater,  and  in  the  ratio  of  the  masses  moved.  The 
expression  for  the  velocity  of  m',  which  would  be  generated  in  a  unit 
of  time  if  the  force  remained  constant,  is  obtained  by  dividing  the 
absolute  force  exerted  by  m  by  the  mass  moved,  which  gives 


.    7 

and  this  is,  therefore,  the  measure  of  the  acceleration  due  to  the 
action  of  m  at  the  distance  p.  For  the  acceleration  due  to  the 
action  of  mr  we  derive,  in  a  similar  manner, 


6.  Observation  shows  that  the  heavenly  bodies  are  nearly  spherical 
in  form,  and  we  shall  therefore,  preparatory  to  finding  the  equations 
which  express  the  relative  motions  of  the  bodies  of  the  system,  de- 
termine the  attraction  of  a  spherical  mass  of  uniform  density,  or 
varying  from  the  centre  to  the  surface  according  to  any  law,  for  a 
point  exterior  to  it. 

If  we  suppose  a  straight  line  to  be  drawn  through  the  centre  of  the 
sphere  and  the  point  attracted,  the  total  action  of  the  sphere  on  the 
point  will  be  a  force  acting  along  this  line,  since  the  mass  of  the 
sphere  is  symmetrical  with  respect  to  it.  Let  dm  denote  an  element 


20  THEORETICAL   ASTRONOMY. 

of  the  mass  of  the  sphere,  and  p  its  distance  from  the  point  attracted  ; 
then  will 

dm 


express  the  action  of  this  element  on  the  point  attracted.  If  we  sup- 
pose the  density  of  the  sphere  to  be  constant,  and  equal  to  unity,  the 
element  dm  becomes  an  element  of  volume,  and  will  be  expressed  by 

dm  =  dx  dy  dz  ; 

xy  y.  and  z  being  the  co-ordinates  of  the  element  referred  to  a  system 
of  rectangular  co-ordinates.  If  we  take  the  origin  of  co-ordinates 
at  the  centre  of  the  sphere,  and  introduce  polar  co-ordinates,  so  that 

x  =  r  cos  <p  cos  0, 
y  —  r  cos  (p  sin  0, 
z  =  r  sin  <p, 

the  expression  for  dm  becomes 

dm  —  r2  cos  <p  dr  dy  do  ; 
and  its  action  on  the  point  attracted  is 

,f  _  r2  cos  <p  dr  d<p  do 

*-      —jf- 

If  we  suppose  the  axis  of  z  to  be  directed  to  the  point  attracted, 
the  co-ordinates  of  this  point  will  be 


a  being  the  distance  of  the  point  from  the  centre  of  the  sphere,  and, 
since 

P*=(x-  xj  +  (y  -  </)2  +  (z  -  *')', 
we  shall  have 

p*  =  a?  —  2ar  sin  <p  -\-  r2. 

The  component  of  the  force  df  in  the  direction  of  the  line  a,  join- 
ing the  point  attracted  and  the  centre  of  the  sphere,  is 

df  cos  Y) 

where  f  is  the  angle  at  the  point  attracted  between  the  element  dm 
and  the  centre  of  the  sphere.  It  is  evident  that  the  sum  of  all  the 
components  which  act  in  the  direction  of  the  line  a  will  express  the 
total  action  of  the  sphere,  since  the  sum  of  those  which  act  perpen- 


ATTRACTION   OF   SPHERES.  21 

dicular  to  this  line,  taken  so  as  to  include  the  entire  mass  of  the 
sphere,  is  zero. 
But  we  have 

a  =  z  -j-  p  cos  Y, 
and  hence 

a  —  r  sin  <p 

cos  Y  = -. 

P 

The  differentiation  of  the  expression  for  />*,  with  respect  to  a,  gives 

dp       a  —  r  sin  <p 

-y-  = =  cos  Y. 

da  p 

Therefore,  if  we  denote  the  attraction  of  the  sphere  by  A,  we  shall 
have,  by  means  of  the  values  of  df  and  cos  r, 

,  .       r2  cos  (p  dr  dy  dd    dp 

a  A.  = .  —=-, 

P*  da 

or 

dA  =  —  r2  cos  <p  dr  d<p  do  --=-. 
da 

The  polar  co-ordinates  r,  <p,  and  0  are  independent  of  a,  and  hence 

d- 


dA  =  — 


p 


da 
Let  us  now  put 

,  Tr     r2  cos  <p  dr  dy  do 


and  we  shall  have 


A-      *? 
•**•  —     "  j  * 

da 


Consequently,  to  find  the  total  action  of  the  sphere  on  the  given 
point,  we  have  only  to  find  V  by  means  of  equation  (2),  the  limits 
of  the  integration  being  taken  so  as  to  include  the  entire  mass  of  the 
sphere,  and  then  find  its  differential  coefficient  with  respect  to  a. 

If  we  integrate  equation  (2)  first  with  reference  to  d,  for  which  p 
is  constant,  between  the  limits  0  =  0  and  d  =  2x9  we  get 

cos  <f>  dr  d<f> 


P 
Thia  must  be  integrated  between  the  limits  tp  =  -f-  fa  an(^  ^  = 


22  THEORETICAL   ASTRONOMY. 

but  since  p  is  a  function  of  <p,  if  we  differentiate  the  expression  for 
p2  with  respect  to  ^>,  we  have 

7  P       7 

r  cos  <p  d<p  = dp, 

a 

and  hence 


Corresponding  to  the  limits  of  (p  we  have  p  =  a  —  r,  and  p  =  a  -f  r; 
and  taking  the  integral  with  respect  to  p  between  these  limits,  we 
obtain 

F-  —  />*•. 

a  J 
Integrating,  finally,  between  the  limits  r  —  0  and  r  =  r,,  we  get 


r,  being  the  radius  of  the  sphere,  and,  if  we  denote  its  entire  mass  by 
m,  this  becomes 


F=-. 
a' 


Therefore, 


dV     m 


da       a? 


from  which  it  appears  that  the  action  of  a  homogeneous  spherical 
mass  on  a  point  exterior  to  it,  is  the  same  as  if  the  entire  mass  were 
concentrated  at  its  centre.  If,  in  the  integration  with  respect  to  r, 
we  take  the  limits  r'  and  r",  we  obtain 


and,  denoting  by  m0  the  mass  of  a  spherical  shell  whose  radii  are  r" 
and  rf,  this  becomes 


Consequently,  the  attraction  of  a  homogeneous  spherical  shell  on  a 
point  exterior  to  it,  is  the  same  as  if  the  entire  mass  were  concentrated 
at  its  centre. 

The  supposition   that  the  point  attracted   is   situated   within  a 
spherical  shell  of  uniform  density,  does  not  change  the  form  of  the 


FUNDAMENTAL    PRINCIPLES. 


general  equation;  but,  in  the  integration  with  reference  to  />,  the 
limits  will  be  p  =  r  +  a,  and  p  =  r  —  a,  which  give 


V=  — 

and  this  being  independent  of  a,  we  have 

A-    -d-?=0 

^i-    -  7  -    \Jm 

da 

Whence  it  follows  that  a  point  placed  in  the  interior  of  a  spherical 
shell  is  equally  attracted  in  all  directions,  and  that,  if  not  subject  to 
the  action  of  any  extraneous  force,  it  will  be  in  equilibrium  in  every 
position. 

7.  Whatever  may  be  the  law  of  the  change  of  the  density  of  the 
heavenly  bodies  from  the  surface  to  the  centre,  we  may  regard  them 
as  composed  of  homogeneous,  concentric  layers,  the  density  varying 
only  from  one  layer  to  another,  and  the  number  of  the  layers  may 
be  indefinite.  The  action  of  each  of  these  will  be  the  same  as  if  its 
mass  were  united  at  the  centre  of  the  shell  ;  and  hence  the  total  action 
of  the  body  will  be  the  same  as  if  the  entire  mass  were  concentrated 
at  its  centre  of  gravity.  The  planets  are  indeed  not  exactly  spheres, 
but  oblate  spheroids  differing  but  little  from  spheres  ;  and  the  error 
of  the  assumption  of  an  exact  spherical  form,  so  far  as  it  relates  to 
their  action  upon  each  other,  is  extremely  small,  and  is  in  fact  com- 
pensated by  the  magnitude  of  their  distances  from  each  other  ;  for, 
whatever  may  be  the  form  of  the  body,  if  its  dimensions  are  small 
in  comparison  with  its  distance  from  the  body  which  it  attracts,  it  is 
evident  that  its  action  will  be  sensibly  the  same  as  if  its  entire  mass 
were  concentrated  at  its  centre  of  gravity.  If  we  suppose  a  system 
of  bodies  to  be  composed  of  spherical  masses,  each  unattended  with 
any  satellite,  and  if  we  suppose  that  the  dimensions  of  the  bodies 
are  small  in  comparison  with  their  mutual  distances,  the  formation 
of  the  equations  for  the  motion  of  the  bodies  of  the  system  will  be 
reduced  to  the  consideration  of  the  motions  of  simple  points  endowed 
with  forces  of  attraction  corresponding  to  the  respective  masses.  Our 
solar  system  is,  in  reality,  a  compound  system,  the  several  systems 
of  primary  and  satellites  corresponding  nearly  to  the  case  supposed  ; 
and,  before  proceeding  with  the  formation  of  the  equations  which  are 
applicable  to  the  general  case,  we  will  consider,  at  first,  those  for  a 
simple  system  of  bodies,  considered  as  points  and  subject  to  their 
mutual  actions  and  the  action  of  the  forces  which  correspond  to  the 


24  THEORETICAL   ASTRONOMY. 

actual  velocities  of  the  different  parts  of  the  system  for  any  instant. 
It  is  evident  that  we  cannot  consider  the  motion  of  any  single  body 
as  free,  and  subject  only  to  the  action  of  the  primitive  impulsion 
which  it  has  received  and  the  accelerating  forces  which  act  upon  it  ; 
but,  on  the  contrary,  the  motion  of  each  body  will  depend  on  the 
force  which  acts  upon  it  directly,  and  also  on  the  reaction  due  to  the 
other  bodies  of  the  system.  The  coLsideration,  however,  of  the  varia- 
tions of  the  motion  of  the  several  bodies  of  the  system  is  reduced  to 
the  simple  case  of  equilibrium  by  means  of  the  general  principle  that, 
if  we  assign  to  the  different  bodies  of  the  system  motions  which  are 
modified  by  their  mutual  action,  we  may  regard  these  motions  as 
composed  of  those  which  the  bodies  actually  have  and  of  other 
motions  which  are  destroyed,  and  which  must  therefore  necessarily 
be  such  that,  if  they  alone  existed,  the  system  would  be  in  equi- 
librium. We  are  thus  enabled  to  form  at  once  the  equations  for  the 
motion  of  a  system  of  bodies.  Let  m,  m',  m",  &c.  be  the  masses  of 
the  several  bodies  of  the  system,  and  x,  y,  z,  x't  y',  zf,  &c.  their  co- 
ordinates referred  to  any  system  of  rectangular  axes.  Further,  let 
the  components  of  the  total  force  acting  upon  a  unit  of  the  mass  of 
m,  or  of  the  accelerating  force,  resolved  in  directions  parallel  to  the 
co-ordinate  axes,  be  denoted  by  X,  F,  and  Z,  respectively,  then  will 

mX,  m  Y,  mZ, 

be  the  forces  which  act  upon  the  body  in  the  same  directions.  The 
velocities  of  the  body  m  at  any  instant,  in  directions  parallel  to  the 

co-ordinate  axes,  will  be 

\ 

dx  dy  dz 

~di'  W  dt' 

and  the  corresponding  forces  are 

dx  dy  dz 

m  ~j7>  m  ~jl*  m  ~jT' 

dt  dt  dt 

By  virtue  of  the  action  of  the  accelerating  force,  these  forces  for  the 
next  instant  become 


*j  ••+  mXdt,  .m+mYdt,  mj  +  mZdt> 

which  may  be  written  respectively: 


MOTION   OF   A   SYSTEM   OF   BODIES.  25 


dz  ,  dz  ..  dz 


The  actual  velocities  for  this  instant  are 

dx        ,  dx  dy          dy  dz  dz 


and  the  corresponding  forces  are 

dx  ,  dx  dy  dy  dz    .       .  dz 

-  - 


Comparing  these  with  the  preceding  expressions  for  the  forces,  it 
appears  that  the  forces  which  are  destroyed,  in  directions  parallel  to 
the  co-ordinate  axes,  are 

dx 

—  md  -j-  -|-  mXdt, 

-md^  +  mYdt,  (3) 

—  md-j--{-mZdt. 

In  the  same  manner  we  find  for  the  forces  which  will  be  destroyed 
in  the  case  of  the  body  m'  : 


-m'd       +  m'X'dt, 
at 


and  similarly  for  the  other  bodies  of  the  system.  According  to  the 
general  principle  above  enunciated,  the  system  under  the  action  of 
these  forces  alone,  will  be  in  equilibrium.  The  conditions  of  equi- 
librium for  a  system  of  points  of  invariable  but  arbitrary  form,  and 
subject  to  the  action  of  forces  directed  in  any  manner  whatever,  are 

ZX,  =  0,  ZY,  =  0,  ZZ,  =  0, 

S  ( T>  -  X,y)  =  0,         Z  (Xtz  -  Z,x)  =  0,         Z  (Z,y  -  I»  =  0 , 

which  X,,  Yh  Z,,  denote  the  components,  resolved  parallel  to  the 


n 


26  THEORETICAL   ASTRONOMY. 

co-ordinate  axes,  of  the  forces  acting  on  any  point,  and  r,  y,  z,  the 
co-ordinates  of  the  point.  These  equations  are  equally  applicable  to 
the  case  of  the  equilibrium  at  any  instant  of  a  system  of  variable 
form ;  and  substituting  in  them  the  expressions  (3)  for  the  forces  de- 
stroyed in  the  case  of  a  system  of  bodies,  we  shall  have 

2m  — r-5  —  2mX  =  0, 
aP 


(4) 


which  are  the  general  equations  for  the  motions  of  a  system  of  bodies. 

8.  Let  x,,  y,,  z,,  be  the  co-ordinates  of  the  centre  of  gravity  of  the 
system,  and,  by  differentiation  of  the  equations  for  the  co-ordinates 
of  the  centre  of  gravity,  which  are 


2mx  2my  2mz 

x,  =  —^  —  )       y.  =  —  ^  —  t       z.  =  —^  —  > 
2m  2m  2m 


we  get 


d*    2m       dP    2m 


Introducing  these  values  into  the  first  three  of  equations  (4),  they 

become 

d*xf 2mX  d*y, 2m  Y  d*zf 2mZ  t  . 

"dP  =  ''  ~^y  ~dP  ~ '  ~2m'  ~dP  ~  ~2m  ' 

from  which  it  appears  that  the  centre  of  gravity  of  the  system  moves 
in  space  as  if  the  masses  of  the  different  bodies  of  which  it  is  com- 
posed, were  united  in  that  point,  and  the  forces  directly  applied  to  it. 
If  we  suppose  that  the  only  accelerating  forces  which  act  on  the 
bodies  of  the  system,  are  those  which  result  from  their  mutual  action, 
we  have  the  obvious  relation : 

=  —  m'X', 


MOTION   OF   A   SYSTEM   OF   BODIES.  27 

and  similarly  for  ^any  two  bodies  ;  and,  consequently, 


so  that  equations  (5)  become 


Integrating  these  once,  and  denoting  the  constants  of  integration  by 
c,  c',  c",  we  find,  by  combining  the  results, 


and  hence  the  absolute  motion  of  the  centre  of  gravity  of  the  system, 
when  subject  only  to  the  mutual  action  of  the  bodies  which  compose 
it,  must  be  uniform  and  rectilinear.  Whatever,  therefore,  may  be 
the  relative  motions  of  the  different  bodies  of  the  system,  the  motion 
of  its  centre  of  gravity  is  not  thereby  affected. 

9.  Let  us  now  consider  the  last  three  of  equations  (4),  and  suppose 
the  system  to  be  submitted  only  to  the  mutual  action  of  the  bodies 
which  compose  it,  and  to  a  force  directed  toward  the  origin  of  co- 
ordinates. The  action  of  m'  on  ra,  according  to  the  law  of  gravita- 

tion, is  expressed  by  —  ,  in  which  p  denotes  the  distance  of  m  from  m'. 

To  resolve  this  force  in  directions  parallel  to  the  three  rectangular 
axes,  we  must  multiply  it  by  the  cosine  of  the  angle  which  the  line 
joining  the  two  bodies  makes  with  the  co-ordinate  axes  respectively, 
which  gives 


m'(z'-z) 
~~ 


Further,  for  the  components  of  the  accelerating  force  of  m  on  m',  we 
have 

m(x  —  x')  ,      m(y  —  yf)  „,       m(g  —  /) 

""?"'  ~7  ~7~ 

Hence  we  derive 

m  (  Yx  —  Xy}  +  m'  (  TV  —  Xy  )  =  0, 
and  generally 


28  THEORETICAL   ASTRONOMY. 

In  a  similar  manner,  we  find 

Zm  (Xi  —  ZOD)  =  0,  (7) 

Im  (Zq  —  F«)  =  0. 

These  relations  will  not  be  altered  if,  in  addition  to  their  reciprocal 
action,  the  bodies  of  the  system  are  acted  upon  by  forces  directed  to 
the  origin  of  co-ordinates.  Thus,  in  the  case  of  a  force  acting  upon 
m,  and  directed  to  the  origin  of  co-ordinates,  we  have,  for  its  action 
alone, 

Yx  =  Xy,  Xz  =  Zx,  %y=Yz, 

and  similarly  for  the  other  bodies.  Hence  these  forces  disappear 
from  the  equations,  and,  therefore,  when  the  several  bodies  of  the 
system  are  subject  only  to  their  reciprocal  action  and  to  forces  directed 
to  the  origin  of  co-ordinates,  the  last  three  of  equations  (4)  become 


d*z 

^ 

d*y 


the  integration  of  which  gives 


2m  {xdy  —  ydx)  =  cat, 

Im  (zdx  —  xdz)  =  c'dt,  (8; 

2m  (ydz  —  zdy]  =  c"dt, 

c,  c',  and  G"  being  the  constants  of  integration.  Now,  xdy  —  ydx  is 
double  the  area  described  about  the  origin  of  co-ordinates  by  the  pro- 
jection of  the  radius-  vector,  or  line  joining  m  with  the  origin  of  co-ordi- 
nates, on  the  plane  of  xy  during  the  element  of  time  dt  ;  and,  further, 
zdx  —  xdz  and  ydz  —  zdy  are  respectively  double  the  areas  described, 
during  the  same  time,  by  the  projection  of  the  radius-vector  on  the 
planes  of  xz  and  yz.  The  constant  c,  therefore,  expresses  double  the 
sum  of  the  products  formed  by  multiplying  the  areal  velocity  of  each 
body,  in  the  direction  of  the  co-ordinate  plane  xy,  by  its  mass;  and 
cr,  Gfr,  express  the  same  sum  with  reference  to  the  co-ordinate  planes 
xz  and  yz  respectively.  Hence  the  sum  of  the  areal  velocities  of  the 
several  bodies  of  the  system  about  the  origin  of  co-ordinates,  each 
multiplied  by  the  corresponding  mass,  is  constant;  and  the  sum  of 
the  areas  traced,  each  multiplied  by  the  corresponding  mass,  is  pro- 
portional to  the  time.  If  the  only  forces  which  operate,  are  those 


INVARIABLE   PLANE.  29 

resulting  from  the  mutual  action  of  the  bodies  which  compose  the 
system,  this  result  is  correct  whatever  may  be  the  point  in  space 
taken  as  the  origin  of  co-ordinates. 

The  areas  described  by  the  projection  of  the  radius-vector  of  each 
body  on  the  co-ordinate  planes,  are  the  projections,  on  these  planes,  of 
the  areas  actually  described  in  space.  We  may,  therefore,  conceive  of 
a  resultant,  or  principal  plane  of  projection,  such  that  the  sum  of  the 
areas  traced  by  the  projection  of  each  radius-vector  on  this  plane, 
when  projected  on  the  three  co-ordinate  planes,  each  being  multiplied 
by  the  corresponding  mass,  will  be  respectively  equal  to  the  first 
members  of  the  equations  (8).  Let  «,  /9,  and  7-  be  the  angles  which 
this  principal  plane  makes  with  the  co-ordinate  planes  xy,  xz,  and  yz, 
respectively;  and  let  S  denote  the  sum  of  the  areas  traced  on  this 
plane,  in  a  unit  of  time,  by  the  projection  of  the  radius-vector  of 
each  of  the  bodies  of  the  system,  each  area  being  multiplied  by  the 
corresponding  mass.  The  sum  S  will  be  found  to  be  a  maximum, 
and  its  projections  on  the  co-ordinate  planes,  corresponding  to  the 
element  of  time  dt,  are 

S  cos  a  dt,  S  cos  p  dt,  S  cos  Y  dt. 

Therefore,  by  means  of  equations  (8),  we  have 

c  =  S  cos  a,  cf  —  S  cos  /?,  c"  =  S  cos  p, 

and,  since  cos2  a  -f-  cos2/?  +  cos2f  =  1, 

£2  =  c2-f  c'2-f  c"2. 


Hence  we  derive 

c' 


COS  a  = 


1/V  +  c'2  +  c"2  1/c2  +  c'2  -j-  c"2 

c" 

cos  y  =  — - 


These  angles,  being  therefore  constant  and  independent  of  the  time, 
show  that  this  principal  plane  of  projection  remains  constantly  par- 
allel to  itself  during  the  motion  of  the  system  in  space,  whatever 
may  be  the  relative  positions  of  the  several  bodies;  and  for  this 
reason  it  is  called  the  invariable  plane  of  the  system.  Its  position 
with  reference  to  any  known  plane  is  easily  determined  when  the 
velocities,  in  directions  parallel  to  the  co-ordinate  axes,  and  the 
masses  and  co-ordinates  of  the  several  bodies  of  the  system,  are 
known.  The  values  of  c,  c',  c"  are  given  by  equations  (8),  and 


THEORETICAL   ASTRONOMY. 

hence  the  values  of  «,  /?,  and  ?-,  which  determine  the  position  of  the 
invariable  plane. 

Since  the  positions  of  the  co-ordinate  planes  are  arbitrary,  we  may 
suppose  that  of  xy  to  coincide  with  the  invariable  plane,  which  gives 
cos  /9  =  0  and  cos  p  =  0,  and,  therefore,  c'  —  0  and  c"  =  0.  Further, 
since  the  positions  of  the  axes  of  x  and  y  in  this  plane  are  arbitrary, 
it  follows  that  for  every  plane  perpendicular  to  the  invariable  plane, 
the  sum  of  the  areas  traced  by  the  projections  of  the  radii-vectores 
of  the  several  bodies  of  the  system,  each  multiplied  by  the  corre- 
sponding mass,  is  zero.  It  may  also  be  observed  that  the  value  of  8 
is  constant  whatever  may  be  the  position  of  the  co-ordinate  planes, 
and  that  its  value  is  necessarily  greater  than  that  of  either  of  the 
quantities  in  the  second  member  of  the  equation, 

£2  =  c2-f  c'2-f  c"2, 

except  when  two  of  them  are  each  equal  to  zero.  It  is,  therefore,  a 
maximum,  and  the  invariable  plane  is  also  the  plane  of  maximum 
areas. 

10.  If  we  suppose  the  origin  of  co-ordinates  itself  to  move  with 
uniform  and  rectilinear  motion  in  space,  the  relations  expressed  by 
equations  (8)  will  remain  unchanged.  Thus,  let  x,,  yn  z,  be  the  co- 
ordinates of  the  movable  origin  of  co-ordinates,  referred  to  a  fixed 
point  in  space  taken  as  the  origin  ;  and  let  XQ,  y0,  z0,  x0f,  yj,  z/,  &c. 
be  the  co-ordinates  of  the  several  bodies  referred  to  the  movable 
origin.  Then,  since  the  co-ordinate  planes  in  one  system  remain 
always  parallel  to  those  of  the  other  system  of  co-ordinates,  we  shall 
have 


and  similarly  for  the  other  bodies  of  the  system.     Introducing  these 
values  of  x,  y,  and  z  into  the  first  three  of  equations  (4),  they  become 


The  condition  of  uniform  rectilinear  motion  of  the  movable  origin 
gives 


MOTION   OF   A   SOLID   BODY.  31 

and  the  preceding  equations  become 


0,  (9) 

—  ZmZ=Q. 


Substituting  the  same  values  in  the  last  three  of  equations  (4),  ob- 
serving that  the  co-ordinates  xt,  y,,  z,  are  the  same  for  all  the  bodies 
of  the  system,  and  reducing  the  resulting  equations  by  means  of 
equations  (9),  we  get 


(/72r  f]i~     \ 

*.  ^jr  ~  *»~jjr  }  -  a*  (A  -  2O  =  0,  (10) 


-  r    =0. 


Hence  it  appears  that  the  form  of  the  equations  for  the  motion  of  the 
system  of  bodies,  remains  unchanged  when  we  suppose  the  origin  of 
co-ordinates  to  move  in  space  with  a  uniform  and  rectilinear  motion. 

11.  The  equations  already  derived  for  the  motions  of  a  system  of 
bodies,  considered  as  reduced  to  material  points,  enable  us  to  form  at 
once  those  for  the  motion  of  a  solid  body.  The  mutual  distances  of 
the  parts  of  the  system  are,  in  this  case,  invariable,  and  the  masses 
of  the  several  bodies  become  the  elements  of  the  mass  of  the  solid 
body.  If  we  denote  an  element  of  the  mass  by  c?m,  the  equations  (5) 
for  the  motion  of  the  centre  of  gravity  of  the  body  become 


-=zdm,     (11) 


the  summation,  or  integration  with  reference  to  dm,  being  taken  so  as 
to  include  the  entire  mass  of  the  body,  from  which  it  appears  that 
the  centre  of  gravity  of  the  body  moves  in  space  as  if  the  entire  mass 
were  concentrated  in  that  point,  and  the  forces  applied  to  it  directly. 
If  we  take  the  origin  of  co-ordinates  at  the  centre  of  gravity  of 
the  body,  and  suppose  it  to  have  a  rectilinear,  uniform  motion  in 
space,  and  denote  the  co-ordinates  of  the  element  dm,  in  reference  to 
this  origin,  by  x0,  yQ)  zw  we  have,  by  means  of  the  equations  (10), 


THEORETICAL   ASTRONOMY. 
&XR 


d*y0  d\  \  .          /-,__          _  N  ,          .v 

^  ~dl y°  ~di?  }       ~~J  '      °  ~~     y*)        ~   ' 

d'2xn  d*z    \  r 

z0-^-~xo-^-Jdm—J(Xz0—Zx0)dm  =  0, 

d\\,         Crr 
-%-$-  )dm-J(Zy0~Yz0)dm  =0, 


0  dt2 

the  integration  with  respect  to  dm  being  taken  so  as  to  include  the 
entire  mass  of  the  body.  These  equations,  therefore,  determine  the 
motion  of  rotation  of  the  body  around  its  centre  of  gravity  regarded 
as  fixed,  or  as  having  a  uniform  rectilinear  motion  in  space.  Equa- 
tions (11)  determine  the  position  of  the  centre  of  gravity  for  any 
instant,  and  hence  for  the  successive  instants  at  intervals  equal  to  dt; 
and  we  may  consider  the  motion  of  the  body  during  the  element  of 
time  dt  as  rectilinear  and  uniform,  whatever  may  be  the  form  of  its 
trajectory.  Hence,  equations  (11)  and  (12)  completely  determine  the 
position  of  the  body  in  space, — the  former  relating  to  the  motion  of 
translation  of  the  centre  of  gravity,  and  the  latter  to  the  motion  of 
rotation  about  this  point.  It  follows,  therefore,  that  for  any  forces 
which  act  upon  a  body  we  can  always  decompose  the  actual  motion 
into  those  of  the  translation  of  the  centre  of  gravity  in  space,  and  of 
the  motion  of  rotation  around  this  point;  and  these  two  motions  may 
be  considered  independently  of  each  other,  the  motion  of  the  centre 
of  gravity  being  independent  of  the  form  and  position  of  the  body 
about  this  point. 

If  the  only  forces  which  act  upon  the  body  are  the  reciprocal  action 
0f  the  elements  of  its  mass  and  forces  directed  to  the  origin  of  co- 
ordinates, the  second  terms  of  equations  (12)  become  each  equal  to 
zero,  and  the  results  indicated  by  equations  (8)  apply  in  this  case 
also.  The  parts  of  the  system  being  invariably  connected,  the  plane 
of  maximum  areas,  or  invariable  plane,  is  evidently  that  which  is 
perpendicular  to  the  axis  of  rotation  passing  through  the  centre  of 
gravity,  and  therefore,  in  the  motion  of  translation  of  the  centre  of 
gravity  in  space,  the  axis  of  rotation  remains  constantly  parallel  to 
itself.  Any  extraneous  force  which  tends  to  disturb  this  relation 
will  necessarily  develop  a  contrary  reaction,  and  hence  a  rotating  body 
resists  any  change  of  its  plane  of  rotation  not  parallel  to  itself.  Wo 
may  observe,  also,  that  on  account  of  the  invariability  of  the  mutual 
distances  of  the  elements  of  the  mass,  according  to  equations  (8),  the 
motion  of  rotation  must  be  uniform. 

12.  We  shall  now  consider  the  action  of  a  svstem  of  bodies  on  a 


MOTION   OF   A  SOLID   BODY.  33 

distant  mass,  which  we  will  denote  by  M.  Let  x0)  yw  zw  x0f,  2/0',  zQ', 
&c.  be  the  co-ordinates  of  the  several  bodies  of  the  system  referred 
to  its  centre  of  gravity  as  the  origin  of  co-ordinates;  xlt  y,,  and  z, 
the  co-ordinates  of  the  centre  of  gravity  of  the  system  referred  to 
the  centre  of  gravity  of  the  body  M.  The  co-ordinates  of  the  body 
m,  of  the  system,  referred  to  this  origin,  will  therefore  be 


and  similarly  for  the  other  bodies  of  the  system.  If  we  denote  by 
r  the  distance  of  the  centre  of  gravity  of  m  from  that  of  My  the 
accelerating  force  of  the  former  on  an  element  of  mass  at  the  centre 
of  gravity  of  the  latter,  resolved  parallel  to  the  axis  of  x,  will  be 


mx 


and,  therefore,  that  of  the  entire  system  on  the  element  of  Mt  resolved 
;n  the  same  direction,  will  be 

vmx 
r5"' 
We  have  also 

r*  =  (x,  +  xQY  +  (y,  +  2/0)2  +  0,  +  *»)', 

and,  if  we  denote  by  rt  the  distance  of  the  centre  of  gravity  of  the 
system  from  Jf, 

r,»  =  *,»  +  ?,«  +  *,*, 

Therefore 

|  =  O,  +  *o)  (r?  +  2  (x,  x0  +  y,  y0  +  z,  $  +  r02) 

We  shall  now  suppose  the  mutual  distances  of  the  bodies  of  the 
system  to  be  so  small  in  comparison  with  the  distance  r,  of  its  centre 
of  gravity  from  that  of  Jf,  that  terms  of  the  order  r02  may  be  neglected  ; 
a  condition  which  is  actually  satisfied  in  the  case  of  the  secondary 
systems  belonging  to  the  solar  system.  Hence,  developing  the  second 
factor  of  the  second  member  of  the  last  equation,  and  neglecting  terms 
of  the  order  r02,  we  shall  have 

»  _  _£/  ,  ^o  _  3s,  (Xa?o  +  y/ 

r3  „  3      '      „,  3  r  5 

r,        rf  rt 

and 

-mx  -m    .   2mxn       3rc.  ,    „  „ 

r—  ==  x,  -§  +  —3°  —  -i  (x,SmxQ  +  y,ZmyQ 
T  r,  rf  i, 

I 


34  THEORETICAL   ASTRONOMY. 

But,  since  #0,  y0,  z0,  are  the  co-ordinates  in  reference  to  the  centre  of 
gravity  of  the  system  as  the  origin,  we  have 


2mx0  =  0,  ZmyQ  =  0,  2wz0  ==  0, 

and  the  preceding  equation  reduces  to 

vw#  2m 

s-?=*'^f- 

In  a  similar  manner,  we  find 

vmy          Im  mz         Im 

V=JV  r7  =  V 

The  second  members  of  these  equations  are  the  expressions  for  the 
total  accelerating  force  due  to  the  action  of  the  bodies  of  the  system 
on  Mj  resolved  parallel  to  the  co-ordinate  axes  respectively,  when  we 
consider  the  several  masses  to  be  collected  at  the  centre  of  gravity 
of  the  system.  Hence  we  conclude  that  when  an  element  of  mass 
is  attracted  by  a  system  of  bodies  so  remote  from  it  that  terms  of  the 
order  of  the  squares  of  the  co-ordinates  of  the  several  bodies,  referred 
to  the  centre  of  gravity  of  the  system  as  the  origin  of  co-ordinates, 
may  be  neglected  in  comparison  with  the  distance  of  the  system  from 
the  point  attracted,  the  action  of  the  system  will  be  the  same  as  if 
the  masses  were  all  united  at  its  centre  of  gravity. 

If  we  suppose  the  masses  m,  m',  m",  &c.  to  be  the  elements  of  the 
mass  of  a  single  body,  the  form  of  the  equations  remains  unchanged; 
and  hence  it  follows  that  the  mass  M  is  acted  upon  by  another  mass, 
or  by  a  system  of  bodies,  as  if  the  entire  mass  of  the  body,  or  of  the 
system,  were  collected  at  its  centre  of  gravity.  It  is  evident,  also, 
that  reciprocally  in  the  case  of  two  systems  of  bodies,  in  which  the 
mutual  distances  of  the  bodies  are  small  in  comparison  with  the 
distance  between  the  centres  of  gravity  of  the  two  systems,  their 
mutual  action  is  the  same  as  if  all  the  several  masses  in  each  system 
were  collected  at  the  common  centre  of  gravity  of  that  system  ;  and 
the  two  centres  of  gravity  will  move  as  if  the  masses  were  thus 
united. 

13.  The  results  already  obtained  are  sufficient  to  enable  us  to  form 
the  equations  for  the  motions  of  the  several  bodies  which  compose  the 
solar  system.  If  these  bodies  were  exact  spheres,  which  could  be 
considered  as  composed  of  homogeneous  concentric  spherical  shells, 
the  density  varying  only  from  one  layer  to  another,  the  action  of 


MOTION   OF   A   SYSTEM   OF   BODIES.  35 

each  on  an  element  of  the  mass  of  another  would  be  the  same  as  if 
the  entire  mass  of  the  attracting  body  were  concentrated  at  its  centre 
of  gravity.  The  slight  deviation  from  this  law,  arising  from  the 
ellipsoidal  form  of  the  heavenly  bodies,  is  compensated  by  the  mag- 
nitude of  their  mutual  distances;  and,  besides,  these  mutual  distances 
are  so  great  that  the  action  of  the  attracting  body  on  the  entire  mass 
of  the  body  attracted,  is  the  same  as  if  the  latter  were  concentrated 
at  its  centre  of  gravity.  Hence  the  consideration  of  the  reciprocal 
action  of  the  single  bodies  of  the  system,  is  reduced  to  that  of  material 
points  corresponding  to  their  respective  centres  of  gravity,  the  masses 
of  which,  however,  are  equivalent  to  those  of  the  corresponding 
bodies.  The  mutual  distances  of  the  bodies  composing  the  secondary 
systems  of  planets  attended  with  satellites  are  so  small,  in  comparison 
with  the  distances  of  the  diiferent  systems  from  each  other  and  from 
the  other  planets,  that  they  act  upon  these,  and  are  reciprocally  acted 
upon,  in  nearly  the  same  manner  as  if  the  masses  of  the  secondary 
systems  were  united  at  their  common  centres  of  gravity,  respectively. 
The  motion  of  the  centre  of  gravity  of  a  system  consisting  of  a 
planet  and  its  satellites  is  not  affected  by  the  reciprocal  action  of  the 
bodies  of  that  system,  and  hence  it  may  be  considered  independently 
of  this  action.  The  difference  of  the  action  of  the  other  planets  on 
a  planet  and  its  satellites  will  simply  produce  inequalities  in  the 
relative  motions  of  the  latter  bodies  as  determined  by  their  mutual 
action  alone,  and  will  not  affect  the  motion  of  their  common  centre 
of  gravity.  Hence,  in  the  formation  of  the  equations  for  the  motion 
of  translation  of  the  centres  of  gravity  of  the  several  planets  or 
secondary  systems  which  compose  the  solar  system,  we  have  simply 
to  consider  them  as  points  endowed  with  attractive  forces  correspond- 
ing to  the  several  single  or  aggregated  masses.  The  investigation 
of  the  motion  of  the  satellites  of  each  of  the  planets  thus  attended, 
forms  a  problem  entirely  distinct  from  that  of  the  motion  of  the 
common  centre  of  gravity  of  such  a  system.  The  COL  ^deration  of 
the  motion  of  rotation  of  the  several  bodies  of  the  solar  system  about 
their  respective  centres  of  gravity,  is  also  independent  of  the  motion 
of  translation.  If  the  resultant  of  all  the  forces  which  act  upon  a 
planet  passed  through  the  centre  of  gravity,  the  motion  of  rotation 
would  be  undisturbed;  and,  since  this  resultant  in  all  cases  very 
nearly  satisfies  this  condition,  the  disturbance  of  the  motion  of  rota- 
tion is  very  slight.  The  inequalities  thus  produced  in  the  motion 
of  rotation  are,  in  fact,  sensible,  and  capable  of  being  indicated  by 
observation,  only  in  the  case  of  the  earth  and  moon.  It  has,  indeed, 


36  THEORETICAL   ASTRONOMY. 

been  rigidly  demonstrated  that  the  axis  of  rotation  of  the  earth  rela- 
tive to  the  body  itself  is  fixed,  so  that  the  poles  of  rotation  and  the 
terrestrial  equator  preserve  constantly  the  same  position  in  reference 
to  the  surface;  and  that  also  the  velocity  of  rotation  is  constant. 
This  assures  us  of  the  permanency  of  geographical  positions,  and, 
in  connection  with  the  fact  that  the  change  of  the  length  of  the 
mean  solar  day  arising  from  the  variation  of  the  obliquity  of  the 
ecliptic  and  in  the  length  of  the  tropical  year,  due  to  the  action  of 
the  sun,  moon,  and  planets  upon  the  earth,  is  absolutely  insensible, 
— amounting  to  only  a  small  fraction  of  a  second  in  a  million  of 
years, — assures  us  also  of  the  permanence  of  the  interval  which  we 
adopt  as  the  unit  of  time  in  astronomical  investigations. 

14.  Placed,  as  we  are,  on  one  of  the  bodies  of  the  system,  it  is 
only  possible  to  deduce  from  observation  the  relative  motions  of  the 
different  heavenly  bodies.  These  relative  motions  in  the  case  of  the 
comets  and  primary  planets  are  referred  to  the  centre  of  the  sun, 
since  the  centre  of  gravity  of  this  body  is  near  the  centre  of  gravity 
of  the  system,  and  its  preponderant  mass  facilitates  the  integration 
of  the  equations  thus  obtained.  In  the  case,  however,  of  the  secondary 
systems,  the  motions  of  the  satellites  are  considered  in  reference  to 
the  centre  of  gravity  of  their  primaries.  We  shall,  therefore,  form 
the  equations  for  the  motion  of  the  planets  relative  to  the  centre  of 
gravity  of  the  sun ;  for  which  it  becomes  necessary  to  consider  more 
particularly  the  relation  between  the  heterogeneous  quantities,  space, 
time,  and  mass,  which  are  involved  in  them.  Each  denomination, 
being  divided  by  the  unit  of  its  kind,  is  expressed  by  an  abstract 
number;  and  hence  it  offers  no  .difficulty  by  its  presence  in  an  equa- 
tion. For  the  unit  of  space  we  may  arbitrarily  take  the  mean  dis- 
tance of  the  earth  from  the  sun.  and  the  mean  solar  day  may  be 
taken  as  the  unit  of  time.  But,  in  order  that  when  the  space  is 
expressed  by  1,  and  the  time  by  1,  the  force  or  velocity  may  also  be 
expressed  by  1,  if  the  unit  of  space  is  first  adopted,  the  relation  of 
the  time  and  the  mass — which  determines  the  measure  of  the  force — - 
will  be  such  that  the  units  of  both  cannot  be  arbitrarily  chosen. 
Thus,  if  we  denote  by  /  the  acceleration  due  to  the  action  of  the 
mass  m  on  a  material  point  at  the  distance  a,  and  by/'  the  accelera- 
tion corresponding  to  another  mass  m'  acting  at  the  same  distance, 
we  have  the  relation 


MOTION   EELATIVE   TO   THE   SUN.  37 

and  hence,  since  the  acceleration  is  proportional  to  the  mass,  it  may 
be  taken  as  the  measure  of  the  latter.  But  we  have,  for  the  measure 
of/, 


Integrating  this,  regarding  /as  constant,  and  the  point  to  move  from 
a  state  of  rest,  we  get 

«  =  i/P.  (13) 

The  acceleration  in  the  case  of  a  variable  force  is,  at  any  instant, 
measured  by  the  velocity  which  the  force  acting  at  that  instant  would 
generate,  if  supposed  to  remain  constant  in  its  action,  during  a  unit 
of  time.  The  last  equation  gives,  when  t  =  1, 

/=2«; 

and  hence  the  acceleration  is  also  measured  by  double  the  space  which 
would  be  described  by  a  material  point,  from  a  state  of  rest,  during 
a  unit  of  time,  the  force  being  supposed  constant  in  its  action  during 
this  time.  In  each  case  the  duration  of  the  unit  of  time  is  involved 
in  the  measure  of  the  acceleration,  and  hence  in  that  of  the  mass  on 
which  the  acceleration  depends;  and  the  unit  of  mass,  or  of  the  force, 
will  depend  on  the  duration  which  is  chosen  for  the  unit  of  time.  In 
general,  therefore,  we  regard  as  the  unit  of  mass  that  which,  acting 
constantly  at  a  distance  equal  to  unity  on  a  material  point  free  to 
move,  will  give  to  this  point,  in  a  unit  of  time,  a  velocity  which, 
if  the  force  ceased  to  act,  would  cause  it  to  describe  the  unit  of  dis- 
tance in  the  unit  of  time. 

Let  the  unit  of  time  be  a  mean  solar  day;  F  the  acceleration  due 
to  the  force  exerted  by  the  mass  of  the  sun  at  the  unit  of  distance; 
and  /the  acceleration  corresponding  to  the  distance  r;  then  will 


and  k2  becomes  the  measure  of  the  mass  of  the  sun.  The  unit  of 
mass  is,  therefore,  equal  to  the  mass  of  the  sun  taken  as  many  times* 
as  k2  is  contained  in  unity.  Hence,  when  we  take  the  mean  solar 
day  as  the  unit  of  time,  the  mass  of  the  sun  is  measured  by  #*;  by 
which  we  are  to  understand  that  if  the  sun  acted  during  a  mean  solar 
day,  on  a  material  point  free  to  move,  at  a  distance  constantly  equal 
to  the  mean  distance  of  the  earth  from  the  sun,  it  would,  at  the  end 
of  that  time,  have  communicated  to  the  point  a  velocity  which,  if 


38  THEORETICAL   ASTRONOMY. 

the  force  did  not  thereafter  act,  would  cause  it  to  describe,  in  a  unu 
of  time,  the  space  expressed  by  ft?. 

The  acceleration  due  to  the  action  of  the  sun  at  the  unit  of  distance 
is  designated  by  F,  since  the  square  root  of  this  quantity  appears 
frequently  in  the  formulae  which  will  be  derived. 

If  we  take  arbitrarily  the  mass  of  the  sun  as  the  unit  of  mass,  the 
unit  of  time  must  be  determined.  Let  t  denote  the  number  of  mean 
solar  days  which  must  be  taken  for  the  unit  of  time  when  the  unit 
of  mass  is  the  mass  of  the  sun.  The  space  which  the  force  due  to 
this  mass,  acting  constantly  on  a  material  point  at  a  distance  equal  to 
the  mean  distance  of  the  earth  from  the  sun,  would  cause  the  point 
to  describe  in  the  time  t,  is,  according  to  equation  (13), 


But,  since  t  expresses  the  number  of  mean  solar  days  in  the  unit  of 
time,  the  measure  of  the  acceleration  corresponding  to  this  unit  is  2s, 
and  this  being  the  unit  of  force,  we  have 

*V=1; 

and  hence 


Therefore,  if  the  mass  of  the  sun  is  regarded  as  the  unit  of  mass,  the 
number  of  mean  solar  days  in  the  unit  of  time  will  be  equal  to  unity 
divided  by  the  square  root  of  the  acceleration  due  to  the  force  exerted 
by  this  mass  at  the  unit  of  distance.  The  numerical  value  of  k  will 
be  subsequently  found  to  be  0.0172021,  which  gives  58.13244  mean 
solar  days  for  the  unit  of  time,  when  the  mass  of  the  sun  is  taken  as 
the  unit  of  mass. 

15.  Let  x,  y,  z  be  the  co-ordinates  of  a  heavenly  body  referred  to 
the  centre  of  gravity  of  the  sun  as  the  origin  of  co-ordinates;  r  its 
radius-vector,  or  distance  from  this  origin;  and  let  m  denote  the 
quotient  obtained  by  dividing  its  mass  by  that  of  the  sun;  then, 
taking  the  mean  solar  day  as  the  unit  of  time,  the  mass  of  the  sun  is 
expressed  by  F,  and  that  of  the  planet  or  comet  by  W&2.  For  a 
second  body  let  the  co-ordinates  be  xf,  yf,  zf  ;  the  distance  from  the 
sun,  r';  and  the  mass,  ra'F;  and  similarly  for  the  other  bodies  of  the 
system.  Let  the  co-ordinates  of  the  centre  of  gravity  of  the  sun 
referred  to  any  fixed  point  in  space  be  £,  77,  £,  the  co-ordinate  planes 
being  parallel  to  those  of  x,  y,  and  z,  respectively;  then  will  the 


MOTION   RELATIVE   TO   THE   SUN.  39 

mk* 
acceleration  due  to  the  action  of  m  on  the  sun  be  expressed  by  —  . 

and  the  three  components  of  this  force  in  directions  parallel  to  the 
co-ordinate  axes,  respectively,  will  be 


,  , 

r3  r3  r3 

The  action  of  ra'  on  the  sun  will  be  expressed  by 


and  hence  the  acceleration  due  to  the  combined  and  simultaneous 
action  of  the  several  bodies  of  the  system  on  the  sun.  resolved  par- 
allel to  the  co-ordinate  axes,  will  be 


The  motion  of  the  centre  of  gravity  of  the  sun,  relative  to  the  fixed 
origin,  will,  therefore,  be  determined  by  the  equations 


= 


=  «,  =«         (14) 


Let  p  denote  the  distance  of  m  from  mf  ;  pr  its  distance  from  w", 
adding  an  accent  for  each  successive  body  considered;  then  will  the 
action  of  the  bodies  m',  m",  &c.  on  m  be 


of  which  the  three  components  parallel  to  the  co-ordinate  axes,  re- 
spectively, are 


The  action  of  the  sun  on  m,  resolved  in  the  same  manner,  is  expressed 
by 


which  are  negative,  since  the  force  tends  to  diminish  the  co-ordinates 
x,  y,  and  z.  The  three  components  of  the  total  action  of  the  othei 
bodies  of  the  system  on  m  are,  therefore, 


40  THEOKETICAL   ASTRONOMY. 

Vx    ,    wm'tf  —  x) 
"T5"  ~7        ' 

k*y  ;     X<y—  y) 
- 


and,  since  the  co-ordinates  of  m  referred  to  the  fixed  origin  are 

£  +  #>  ^  +  y,  c  +  2, 

the  equations  which  determine  the  absolute  motion  are 

eP£    .   (?a?       ^  __       m'  <X  —  a;) 
~" 


the  symbol  of  summation  in  the  second  members  relating  simply  to 
the  masses  and  co-ordinates  of  the  several  bodies  which  act  on  m, 

d2^    d?f)  d2ff 

exclusive  of  the  sun.     Substituting  for  -^-,  ——,  and  -  -  their  values 

dt*    dt2  dt* 

given  by  equations  (14),  we  get 


(10) 

d?z  , 


z         .       ,  I  z'  —  z        z'  \ 
—  =  VZm1    —  -  ---  -   - 

i*  \     p3          r3  I 


Since  x,  y,  z  are  the  co-ordinates  of  m  relative  to  the  centre  of  gravity 
of  the  sun,  these  equations  determine  the  motion  of  m  relative  to  that 
point.  The  second  members  may  be  put  in  another  form,  which 
greatly  facilitates  the  solution  of  some  of  the  problems  relating  to 
the  motion  of  m.  Thus,  let  us  put 

m'     II     W+yyH-aA        m"    /  1      xx"+yy"+zz" 


-+  + 

(17) 

and  we  styall  have  for  the  partial  differential  coefficient  of  this  with 
respect  to  x, 


dx         l  +  m         ffdx       r»l  +  m        ?"  dx 


MOTION   RELATIVE   TO   THE  SUN.  41 

But,  since 

we  have 

dp  _        xf  —  x  dp'  _        x" — x 

dx  p  dx  p' 

and  hence  we  derive 

ldQ\         m'     ix'  —  x        x'  \  .      m"    [x"—x       x"  \  . 

I  I  == I  3 f3~  I  H (  ^3 "F  I  ~f"  •  •  •  *&C« 

or 


"We  find,  also,  in  the  same  manner,  for  the  partial  differential  coeffi 
cients  with  respect  to  y  and  2, 


The  equations  (16),  therefore,  become 


It  will  be  observed  that  the  second  members  of  equations  (16)  ex- 
press the  difference  between  the  action  of  the  bodies  ra',  m",  &c.  on 
m  and  on  the  sun,  resolved  parallel  to  the  co-ordinate  axes  respect- 
ively. The  mutual  distances  of  the  planets  are  such  that  these  quan- 
tities are  generally  very  small,  and  we  may,  therefore,  in  a  first 
approximation  to  the  motion  of  m  relative  to  the  sun,  neglect  the 
second  members  of  these  equations;  and  the  integrals  which  may 
then  be  derived,  express  what  is  called  the  undisturbed  motion  of  m. 
By  means  of  the  results  thus  obtained  for  the  several  bodies  succes- 
sively, the  approximate  values  of  the  second  members  of  equations 
(16)  may  be  found,  and  hence  a  still  closer  approximation  to  the 
actual  motion  of  m.  The  force  whose  components  are  expressed  by 
the  second  members  of  these  equations  is  called  the  disturbing  force  ; 


42  THEORETICAL   ASTRONOMY. 

and,  using  the  second  form  of  the  equations,  the  function  J2,  which 
determines  these  components,  is  called  the  perturbing  function.  The 
complete  solution  of  the  problem  is  facilitated  by  an  artifice  of  the 
infinitesimal  calculus,  known  as  the  variation  of  parameters,  or  of 
constants,  according  to  which  the  complete  integrals  of  equations  (16) 
are  of  the  same  form  as  those  obtained  by  putting  the  second  mem- 
bers equal  to  zero,  the  arbitrary  constants,  however,  of  the  latter 
integration  being  regarded  as  variables.  These  constants  of  integra- 
tion are  the  elements  which  determine  the  motion  of  m  relative  to  the 
sun,  and  when  the  disturbing  force  is  neglected  the  elements  are  pure 
constants.  The  variations  of  these,  or  of  the  co-ordinates,  arising 
from  the  action  of  the  disturbing  force  are,  in  almost  all  cases,  very 
small,  and  are  called  the  perturbations.  The  problem  which  first 
presents  itself  is,  therefore,  the  determination  of  all  the  circumstances 
of  the  undisturbed  motion  of  the  heavenly  bodies,  after  which  the 
action  of  the  disturbing  forces  may  be  considered. 

It  may  be  further  remarked  that,  in  the  formation  of  the  preceding 
equations,  we  have  supposed  the  different  bodies  to  be  free  to  move, 
and,  therefore,  subject  only  to  their  mutual  action.  There  are,  in- 
deed, facts  derived  from  the  study  of  the  motion  of  the  comets  which 
seem  to  indicate  that  there  exists  in  space  a  resisting  medium  which 
opposes  the  free  motion  of  all  the  bodies  of  the  system.  If  such  a 
medium  actually  exists,  its  effect  is  very  small,  so  that  it  can  be  sen- 
sible only  in  the  case  of  rare  and  attenuated  bodies  like  the  comets, 
since  the  accumulated  observations  of  the  different  planets  do  not 
exhibit  any  effect  of  such  resistance.  But,  if  we  assume  its  existence, 
it  is  evidently  necessary  only  to  add  to  the  second  members  of  equa- 
tions (16)  a  force  which  shall  represent  the  effect  of  this  resistance,  — 
which,  therefore,  becomes  a  part  of  the  disturbing  force,  —  and  the 
motion  of  m  will  be  completely  determined. 

16.  When  we  consider  the  undisturbed  motion  of  a  planet  or 
comet  relative  to  the  sun,  or  simply  the  motion  of  the  body  relative 
to  the  sun  as  subject  only  to  the  reciprocal  action  of  the  two  bodies, 
the  equations  (16)  become 


(19) 


MOTION   RELATIVE   TO   THE   SUN.  43 

The  equations  for  the  undisturbed  motion  of  a  satellite  relative  to  its 
primary  are  of  the  same  form,  the  value  of  &2,  however,  being  in  this 
case  the  acceleration  due  to  the  force  exerted  by  the  mass  of  the 
primary  at  the  unit  of  distance,  and  m  the  ratio  of  the  mass  of  the 
satellite  to  that  of  the  primary. 

The  integrals  of  these  equations  introduce  six  arbitrary  constants 
of  integration,  which,  when  known,  will  completely  determine  the 
undisturbed  motion  of  m  relative  to  the  sun. 

If  we  multiply  the  first  of  these  equations  by  y,  and  the  second  by 
a?,  and  subtract  the  last  product  from  the  first,  we  shall  find,  by  inte- 

grating the  result, 

xdy  —  ydx 

~ 


c  being  an  arbitrary  constant. 
In  a  similar  manner,  we  obtain 

xdz  —  zdx  _   ,  ydz  —  zdy  _  „ 

~~dT       G'  ~~dt~~ 

If  we  multiply  these  three  equations  respectively  by  z,  —  y,  and  x, 
and  add  the  products,  we  obtain 

cz  —  c'y  -f-  c"x  —  0. 

This,  being  the  equation  of  a  plane  passing  through  the  origin  of 
co-ordinates,  shows  that  the  path  of  the  body  relative  to  the  sun  is  a 
plane  curve,  and  that  the  plane  of  the  orbit  passes  through  the  centre 
of  the  sun. 

Again,  if  we  multiply  the  first  of  equations  (19)  by  2dx,  the  second 
by  2dy,  and  the  third  by  2dz,  take  the  sum  and  integrate,  we  shall 
find 


But,  since  r2  =  a?  -f  f  -f  z2,  we  shall  have,  by  differentiation, 

rdr  =  xdx  -f  ydy  +  zdz. 

Therefore,  introducing  this  value  into  the  preceding  equation,  we  obtain 
dat+dtf  +  d*       2fr(l  +  m)   ,   A^0>  (20) 

h  being  an  arbitrary  constant. 


44  THEORETICAL   ASTRONOMY. 

If  we  add  together  the  squares  of  the  expressions  for  c,  c',  and  c" 
and  put  c2  +  c'2  +  c"2  =  4/2,  we  shall  have 

(x*  -f  f  +  *2)  (cfe*  +  dy*  +  <**2)      0*k  +  ydy  -f 

dz2  <ft2 

or 


d* 

"'-- 


If  we  represent  by  dv  the  infinitely  small  angle  contained  between 
two  consecutive  radii-  vectores  r  and  r  -f-  cZr,  since  dx2  +  c??/2  +  c/2^  is 
the  square  of  the  element  of  path  described  by  the  body,  we  shall 

have 

dx*  +  df  -f-  dz*  =  dr*  -f  rzdv*. 

Substituting  this  value  in  the  preceding  equation,  it  becomes 

r*dv  =  2fdt.  (22) 

The  quantity  r2dv  is  double  the  area  included  by  the  element  of  path 
described  in  the  element  of  time  dt,  and  by  the  radii-  vectores  r  and 
r  -f-  dr;  and/,  therefore,  represents  the  areal  velocity,  which,  being  a 
constant,  shows  that  the  radius-vector  of  a  planet  or  comet  describes 
equal  areas  in  equal  intervals  of  time. 

From  the  equations  (20)  and  (21)  we  find,  by  elimination, 

(23) 


l+m)  —  Ar2—  4/2 
Substituting  this  value  of  dt  in  equation  (22),  we  get 


r  -  (24) 

m)  —  hr*—  4/2 

which  gives,  in  order  to  find  the  maximum  and  minimum  values  of  r, 


dr_  _  r  l2rff  (1  +  m)  —  Ar2  —  4/2  _ 

<fo  "  2/  " 

or 

2rA;2(l  +  m)  —  hr*—  4/2=  C. 
Therefore 


^ h  m)8, 

and 


J       4/2 
\"~" 


4/2 


are,  respectively,  the   maximum   and  minimum  values  of  r.     The 


MOTION   RELATIVE  *TO   THE   SUN.  45 

points  of  the  orbit,  or  trajectory  of  the  body  relative  to  the  sun,  cor- 
responding to  these  values  of  r,  are  called  the  apsides;  the  former, 
the  aphelion,  and  the  latter,  the  perihelion.  If  we  represent  these 
values,  respectively,  by  a  (I  +  «)  and  a(l  —  e),  we  shall  have 

m);  4f  =  ak*  (1  -f-  m)  (1  -  e^  =  Vp  (1  +  m), 


in  which  p  =  a  (1  —  e2).     Introducing  these  values  into  the  equation 
(24),  it  becomes 


^,--,-  -y     ^-(E.i-I)' 

the  integral  of  which  gives 

-1 1  /  P 
v  =  u>  +  cos      —  I 


being  an  arbitrary  constant.     Therefore  we  shall  have 

—  I  —  —  1  1  =  cos  (v  —  w), 
e  \  r          / 


from  which  we  derive 

r  ^^^ 


e  cos  (1; 


which  is  the  polar  equation  of  a  conic  section,  the  pole  being  at  the 
focus,  p  being  the  semi -para  meter,  e  the  eccentricity,  and  v  —  o  the 
angle  at  the  focus  between  the  radius-vector  and  a  fixed  line,  in  the 
plane  of  the  orbit,  making  the  angle  co  with  the  semi-transverse 
axis  a. 

If  the  angle  v  —  co  is  counted  from  the  perihelion,  we  have  a)  =  0, 
and 

(25) 


P 


COS  V 


The  angle  v  is  called  the  true  anomaly. 

Hence  we  conclude  that  the  orbit  of  a  heavenly  body  revolving 
around  the  sun  is  a  conic  section  with  the  sun  in  one  of  the  foci. 
Observation  shows  that  the  planets  revolve  around  the  sun  in  ellipses, 
usually  of  small  eccentricity,  while  the  comets  revolve  either  in 
ellipses  of  great  eccentricity,  in  parabolas,  or  in  hyperbolas,  a  cir- 
cumstance which,  as  we  shall  have  occasion  to  notice  hereafter,  greatly 


46  THEORETICAL   ASTRONOMY. 

lessens  the  amount  of  labor  in  many  computations  respecting  their 
motion. 

Introducing  into  equation  (23)  the  values  of  h  and  4f  already 
found,  we  obtain 


dt  = 


- 
m    j/aV  —  (a  —  r)2' 

which  may  be  written 


dt  = 

or 


^-(V)/ 

the  integration  of  which  gives 
J 


ae 


(26) 


In  the  perihelion,  r  =  a  (1  —  e),  and  the  integral  reduces  to  £'  =  C/ 
therefore,  if  we  denote  the  time  from  the  perihelion  by  tQ,  we  shall 
have 


o          /        -i/a  —  r\  /~       (a  —  r\*\ 

—  7    /  ^r==    cos        -  )  —  e\l—   --       • 
jfe|/l-f-w\  \     ae     j          \          \    ae    ]  ] 


(27) 


In  the  aphelion,  r  =  a  (1  -f-  e)  ;  and  therefore  we  shall  have,  for  the 
time  in  which  the  body  passes  from  the  perihelion  to  the  aphelion, 
t,  =  Jr,  or 

1  T- 

" 

r  being  the  periodic  time,  or  time  of  one  revolution  of  the  planet 
around  the  sun,  a  the  semi-transverse  axis  of  the  orbit,  or  mean  dis- 
tance from  the  sun,  and  n  the  semi-circumference  of  a  circle  whose 
radius  is  unity.  Therefore  we  shall  have 


(28) 


MOTION   RELATIVE   TO   THE   SUN.  47 

For  a  second  planet,  we  shall  have 


and,  consequently,  between  the  mean  distances  and  periodic  times  of 
any  two  planets,  we  have  the  relation 

fft^  =  £-  (29) 


If  the  masses  of  the  two  planets  m  and  mr  are  very  nearly  the 
same,  we  may  take  1  -f-  m  =  1  -f-  mr  •  and  hence,  in  this  case,  it  follows 
that  the  squares  of  the  periodic  times  are  to  each  other  as  the  cubes  of 
the  mean  distances  from  the  sun.  The  same  result  may  be  stated  in 
another  form,  which  is  sometimes  more  convenient.  Thus,  since  rcab 
is  the  area  of  the  ellipse,  a  and  b  representing  the  semi-axes,  we 
shall  have 

—  =f=  areal  velocity; 


and,  since  b2  =  a2  (1  —  e2),  we  have 


T 

which  becomes,  by  substituting  the  value  of  r  already  found, 


In  like  manner,  for  a  second  planet,  we  have 


and,  if  the  masses  are  such  that  we  may  take  1  -f-  m  sensibly  equal 
to  1  -f  m',  it  follows  that,  in  this  case,  the  areas  described  in  equal 
times,  in  different  orbits,  are  proportional  to  the  square  roots  of  their 
parameters. 

17.  We  shall  now  consider  the  signification  of  some  of  the  con- 
stants of  integration  already  introduced.  Let  i  denote  the  inclination 
of  the  orbit  of  m  to  the  plane  of  xy,  which  is  thus  taken  as  the  plane 
of  reference,  and  let  ££  be  the  angle  formed  by  the  axis  of  x  and  the 
line  of  intersection  of  the  plane  of  the  orbit  with  the  plane  of  xy ; 
then  will  the  angles  i  and  &  determine  the  position  of  the  plane  of 


48  THEORETICAL   ASTRONOMY. 

the  orbit  in  space.     The  constants  c,  c',  and  c",  involved  in  the 
equation 

cz  —  c'y  -\-  c"x  =  0, 

are,  respectively,  double  the  projections,  on  the  co-ordinate  planes, 
xy,  xz,  and  yz,  of  the  areal  velocity//  and  hence  we  shall  have 

e  =  2/cosi. 

The  projection  of  2f  on  a  plane  passing  through  the  intersection  ot 
the  plane  of  the  orbit  with  the  plane  of  XT/,  and  perpendicular  to  the 

latter,  is 

2/sini; 

and  the  projection  of  this  on  the  plane  of  xzy  to  which  it  is  inclined 
at  an  angle  equal  to  &,  gives 

c'  =  2/  sin  i  cos  £1. 
Its  projection  on  the  plane  of  yz  gives 


Hence  we  derive 

z  cos  i  —  y  sin  i  cos  Q>  -f-  x  sin  i  sin  ££  =  0,  (31) 

which  is  the  equation  of  the  plane  of  the  orbit;  and,  by  means  of 
the  value  of  f  in  terms  of  p,  and  the  values  of  c,  c',  c",  we  derive, 
also, 


dz         dx  , — 73 — ; T  .     .  ,„,», 

'-j z-j-  =  Jc i/p  (1  +  »*)  cos  &  sm l>  (.&%) 

at          at 


__         =  cvpLm   sn 

Cvv  Ctv 

These  equations  will  enable  us  to  determine  & ,  i,  and  p,  when,  for 
any  instant,  the  mass  and  co-ordinates  of  m,  and  the  components  of 
its  velocity,  in  directions  parallel  to  the  co-ordinate  axes,  are  known. 
The  constants  a  and  e  are  involved  in  the  value  of  p,  and  hence  four 
constants,  or  elements,  are  introduced  into  these  equations,  two  of 
which,  a  and  e,  relate  to  the  form  of  the  orbit,  and  two,  £>  and  i,  to 
the  position  of  its  plane  in  space.  If  we  measure  the  angle  v  —  a) 
from  the  point  in  which  the  orbit  intersects  the  plane  of  xy,  the  con- 
stant co  will  determine  the  position  of  the  orbit  in  its  own  plane. 
Finally,  the  constant  of  integration  C,  in  equation  (26),  is  the  time 


MOTION   RELATIVE   TO   THE   SUN.  4S 

of  passage  through  the  perihelion;  and  this  determines  the  position 
of  the  body  in  its  orbit.  When  these  six  constants  are  known,  the 
undisturbed  orbit  of  the  body  is  completely  determined. 

Let  V  denote  the  velocity  of  the  body  in  its  orbit;  then  will 
equation  (20)  become 


At  the  perihelion,  r  is  a  minimum,  and  hence,  according  to  this 
equation,  the  corresponding  value  of  V  is  a  maximum.  At  the 
aphelion,  V  is  a  minimum. 

In  the  parabola,  a  =  oo*,  and  hence 


which  will  determine  the  velocity  at  any  instant,  when  r  is  known. 
It  will  be  observed  that  the  velocity,  corresponding  to  the  same  value 
of  r,  in  an  elliptic  orbit  is  less  than  in  a  parabolic  orbit,  and  that, 
since  a  is  negative  in  the  hyperbola,  the  velocity  in  a  hyperbolic 
orbit  is  still  greater  than  in  the  case  of  the  parabola.  Further,  since 
the  velocity  is  thus  found  to  be  independent  of  the  eccentricity,  the 
direction  of  the  motion  has  no  influence  on  the  species  of  conic  section 
described. 

If  the  position  of  a  heavenly  body  at  any  instant,  and  the  direction 
and  magnitude  of  its  velocity,  are  given,  the  relations  already  derived 
will  enable  us  to  determine  the  six  constant  elements  of  its  orbit. 
But  since  we  cannot  know  in  advance  the  magnitude  and  direction 
of  the  primitive  impulse  communicated  to  the  body,  it  is  only  by 
the  aid  of  observation  that  these  elements  can  be  derived;  and 
therefore,  before  considering  the  formulae  necessary  to  determine 
unknown  elements  by  means  of  observed  positions,  we  will  investi- 
gate those  which  are  necessary  for  the  determination  of  the  helio- 
centric and  geocentric  places  of  the  body,  assuming  the  elements  to 
be  known.  The  results  thus  obtained  will  facilitate  the  solution  of 
the  problem  of  finding  the  unknown  elements  from  the  data  furnished 
by  observation. 

18.  To  determine  the  value  of  k,  which  is  a  constant  for  the  solar 
system,  we  have,  from  equation  (28), 


50  THEORETICAL   ASTRONOMY. 

In  the  case  of  the  earth,  a  =  1,  and  therefore 

*= 


fl/1  4- 


m 


In  reducing  this  formula  to  numbers  we  should  properly  use,  for  r, 
the  absolute  length  of  the  sidereal  year,  which  is  invariable.  The 
effect  of  the  action  of  the  other  bodies  of  the  system  on  the  earth  is 
to  produce  a  very  small  secular  change  in  its  mean  longitude  correr 
spending  to  any  fixed  date  taken  as  the  epoch  of  the  elements;  and 
a  correction  corresponding  to  this  secular  variation  should  be  applied 
to  the  value  of  r  derived  from  observation.  The  effect  of  this  cor- 
rection is  slightly  to  increase  the  observed  value  of  r;  but  to  deter- 
mine it  with  precision  requires  an  exact  knowledge  of  the  masses  of 
all  the  bodies  of  the  system,  and  a  complete  theory  of  their  relative 
motions, — a  problem  which  is  yet  incompletely  solved.  Astronomical 
usage  has,  therefore,  sanctioned  the  employment  of  the  value  of  k 
found  by  means  of  the  length  of  the  sidereal  year  derived  directly 
from  observation.  This  is  virtually  adopting  as  the  unit  of  space  a 
distance  which  is  very  little  less  than  the  absolute,  invariable  mean 
distance  of  the  earth  from  the  sun;  but,  since  this  unit  may  be  arbi- 
trarily chosen,  the  accuracy  of  the  results  is  not  thereby  affected. 

The  value  of  r  from  which  the  adopted  value  of  k  has  been  com- 
puted, is  365.2563835  mean  solar  days;  and  the  value  of  the  com- 
bined mass  of  the  earth  and  moon  is 


~~  354710* 

Hence  we  have  log  V  =  2.5625978148;  log  j/1  +  m  =  0.0000006122; 
log  2x  =  0.7981798684;  and,  consequently, 

log  k  =  8.2355814414. 

If  we  multiply  this  value  of  k  by  206264.81,  the  number  of  seconds 
of  arc  corresponding  to  the  radius  of  a  circle,  we  shall  obtain  its 
value  expressed  in  seconds  of  arc  in  a  circle  whose  radius  is  unity,  or 
on  the  orbit  of  the  earth  supposed  to  be  circular.  The  value  of  k  in 
seconds  is,  therefore, 

log  k  =  3.5500065746. 

o 

The  quantity  —  expresses  the  mean  angular  motion  of  a  planet 

in  a  mean  solar  day,  and  is  usually  designated  by  //.  "We  shall, 
therefore,  have 


MOTION   RELATIVE   TO   THE   SUN.  51 

^,  '"     '        |     C33) 

for  the  expression  for  the  mean  daily  motion  of  a  planet. 

Since,  in  the  case  of  the  earth,  1/1  -f-  m  differs  very  little  from  1, 
it  will  be  observed  that  k  very  nearly  expresses  the  mean  angular 
motion  of  the  earth  in  a  mean  solar  day. 

In  the  case  of  a  small  planet  or  of  a  comet,  the  mass  m  is  so  small 
that  it  may,  without  sensible  error,  be  neglected  ;  and  then  we  shall 
have 

^  =  4  (M) 

a 

For  the  old  planets  whose  masses  are  considerable,  the  rigorous  ex- 
pression (33)  must  be  used. 

19.  Let  us  now  resume  the  polar  equation  of  the  ellipse,  the  pole 
being  at  the  focus,  which  is 


1  -f  e  cos  v 


If  we  represent  by  <p  the  angle  included  between  the  conjugate  axis 
and  a  line  drawn  from  the  extremity  of  this  axis  to  the  focus,  we 
shall  have 

sin  <f>  =  e; 

and,  since  a(l  —  e2)  is  half  the  parameter  of  the  transverse  axis, 
which  we  have  designated  by  p,  we  have  . 


r  — 


1  -}-  sin  (f>  cos  v 


The  angle  <p  is  called  the  angle  of  eccentricity. 
Again,  since  p  =  a  (I  —  e2)  =  a  cos2  ^,  we  have 


1  -}-  sin  <p  cos  v 


It  is  evident,  from  this  equation,  that  the  maximum  value  of  r  in  an 
elliptic  orbit  corresponds  to  v  =  180°,  and  that  the  minimum  value 
of  r  corresponds  to  v  =  0.  It  therefore  increases  from  the  perihelion 
to  the  aphelion,  and  then  decreases  as  the  planet  approaches  the  peri- 
helion. 


52  THEORETICAL   ASTRONOMY. 

In  the  case  of  the  parabola,  <p  =  90°,  and  sin  <p  =  e  =  1  ;  conse- 
quently, 


. 

1  -j-  COS  V 

But,  since  1  +  cos  v  =  2  cos2  Jt?,  if  we  put  <?  =  £p,  we  shall  have 


(36) 


iii  which  ^  is  the  perihelion  distance.  In  this  case,  therefore,  when 
v  =  rt  180°,  r  will  be  infinite,  and  the  comet  will  never  return,  but 
course  its  way  to  other  systems. 

The  angle  <p  cannot  be  applied  to  the  case  of  the  hyperbola,  since 
in  a  hyperbolic  orbit  e  is  greater  than  1  ;  and,  therefore,  the  eccen- 
tricity cannot  be  expressed  by  the  sine  of  an  arc.  If,  however,  we 
designate  by  4  the  angle  which  the  asymptote  to  the  hyperbola  makes 
with  the  transverse  axis,  we  shall  have 

e  cos  4  =  1. 

Introducing  this  value  of  e  into  the  polar  equation  of  the  hyperbola, 
it  becomes 

p  cos  4 
cosv  -f-  cos  4-' 

But,  since  cos  v  -f  cos  4  =  2  cos  )  (v  +  4)  cos  \(v  —  $),  ^s  giyes 

=  _  ff  cos4  _  ™ 

2  cos  J  (v  +  4)  cos  i  (v  —  4)' 

It  appears  from  this  formula  that  r  increases  with  v,  and  becomes  in- 
finite when  1  +  e  cosv  =  0,  or  cosv  =  —  cost)/,  in  which  case  v  =  180° 
-  4  :  consequently,  the  maximum  positive  value  of  v  is  represented 
by  180°  —  4?  and  the  maximum  negative  value  by  —  (180°  —  4/)« 
Further,  it  is  evident  that  the  orbit  will  be  that  branch  of  the  hyper- 
bola which  corresponds  to  the  focus  in  which  the  sun  is  placed,  since, 
under  the  operation  of  an  attractive  force,  the  path  of  the  body  must 
be  concave  toward  the  centre  of  attraction.  A  body  subject  to  a 
force  of  repulsion  of  the  same  intensity,  and  varying  according  to 
the  same  law,  would  describe  the  other  branch  of  the  curve. 

The  problem  of  finding  the  position  of  a  heavenly  body  as  seen 
from  any  point  of  reference,  consists  of  two  parts:  first,  the  deter- 
mination of  the  place  of  the  body  in  its  orbit;  and  then,  by  means 
if  this  and  of  the  elements  which  fix  the  position  of  the  plane  of  the 


PLACE   IN   THE   CEBIT.  53 

orbit,  and  that  of  the  orbit  in  its  own  plane,  the  determination  of 
the  position  in  space. 

In  deriving  the  formulae  for  finding  the  place  of  the  body  in  its 
orbit,  we  will  consider  each  species  of  conic  section  separately,  com- 
mencing with  the  ellipse. 

20.  Since  the  value  of  a-  —  r  can  never  exceed  the  limits  —  ae  and 
-f  ae,  we  may  introduce  an  auxiliary  angle  such  that  we  shall  have 

a  —  r  ^ 

-  =  cos  E. 
ae 

This  auxiliary  angle  E  is  called  the  eccentric  anomaly  ;  and  its  geo 
metrical  signification  may  be  easily  known  from  its  relation  to  the 

true  anomaly.     Introducing  this  value  of  -        -  into  the  equation 

(27)  and  writing  t  —  T  in  place  of  tQ,  T  being  the  time  of  perihelion 
passage  ,  and  t  the  time  for  which  the  place  of  the  planet  in  its  orbit 
is  to  be  computed,  we  obtain 


3 
a2 


(38) 


I     TTJ, 

But  -  —  -  =  mean  daily  motion  of  the  planet  =  /JL  ;  therefore 


The  quantity  p(k  —  T)  represents  what  would  be  the  angular  distance 
from  the  perihelion  if  the  planet  had  moved  uniformly  in  a  circular 
orbit  whose  radius  is  a,  its  mean  distance  from  the  sun.  It  is  called 
the  mean  anomaly,  and  is  usually  designated  by  M.  We  shall,  there- 

fore, have 

M=v(t-T), 

M=E—esmE.  (39) 

When  the  planet  or  comet  is  in  its  perihelion,  the  true  anomaly, 
mean  anomaly,  and  eccentric  anomaly  are  each  equal  to  zero.  All 
three  of  these  increase  from  the  perihelion  to  the  aphelion,  where 
they  are  each  equal  to  180°,  and  decrease  from  the  aphelion  to  the  peri- 
helion, provided  that  they  are  considered  negative.  From  the  peri- 
helion to  the  aphelion  v  is  greater  than  E,  and  E  is  greater  than  M. 
The  same  relation  holds  true  from  the  aphelion  to  the  perihelion,  if 
WQ  regard,  in  this  case,  the  values  of  v,  E,  and  M  as  negative. 

As  soon  as  the  auxiliary  angle  E  is  obtained  by  means  of  the  mean 
motion  and  eccentricity,  the  values  of  r  and  v  may  be  derived.  For 


54  THEOKETICAL,   ASTEONOMY. 

this  purpose  there  are  various  formulae  which   may  be  applied  in 
practice,  and  which  we  will  now  develop. 
The  equation 

a  —  r 

-  =  cos  E, 
ae 

gives 

r  =  a(l  —  ecosE).  (40) 

This  also  gives 

a  —  r  r, 

--  ae  =  a  cos  E  —  ae, 


or 

P  - 

l 


=  a  cos  JE/  —  ae, 


which,  by  means  of  equation  (25),  reduces  to 

r  cos  v  =  a  cos  E  —  ae.  (41) 

If  we  square  both  members  of  equations  (40)  and  (41),  and  subtract 
the  latter  result  from  the  former,  we  get 

r2sm2v  =  a2(l  —  e2)  sm2^, 
or 


r  sin  v  —  a-j/1  —  e2  sin  E  =  b  sin  E.  (42) 

By  means  of  the  equations  (41)  and  (42)  it  may  be  easily  shown 
that  the  auxiliary  angle  E}  or  eccentric  anomaly,  is  the  angle  at  the 
centre  of  the  ellipse  between  the  semi-transverse  axis,  and  a  line 
drawn  from  the  centre  to  the  point  where  the  prolongation  of  the 
ordinate  perpendicular  to  this  axis,  and  drawn  through  the  place  of 
the  body,  meets  the  circumference  of  the  circumscribed  circle. 

Equations  (40)  and  (41)  give 

r  (1  HH  cos  v)  =  a(l  ±  e)  (1  =F  cos  E). 

By  using  first  the  upper  sign,  and  then  the  lower  sign,  we  obtain,  by 
reduction, 

1/r  sin  £v  —  l/a(l  -j-  e)  sin  \E, 

Vr  cos  ±v  =  Va(l  —  e)  cos  %E,  (43) 

which  are  convenient  for  the  calculation  of  r  and  v,  and  especially  so 
when  several  places  are  required.     By  division,  these  equations  give 


tan  %v  =  \l       ~  tan  1  E.  (44) 


PLACE   IN   THE   ORBIT.  65 

Since  e  =  sin  <p,  we  have 

l-e  =  l^niy  =  ^  _ 

1  -}-  e       1  +  sin  p 


Consequently, 

tan  ±E  =  tan  (45°  —  £?>)  tan  |v.  (45) 

Again. 

V\-\-  e  =  1/1  -f  sin  y  =  1/1  -|-  2  sin  £p  cos  fop, 
which  may  be  written 

1/1  -}-  e  —  I/sin2  -|^  -f  c°s2  i^  +  2  sin  Jp  cos  Jp, 
or  _ 

1/1  -|-  e  =  sin  ^  -f  cos  £?• 

In  a  similar  manner  we  find 

1/1  —  e  =  —  sin  \<p  -j-  cos  \<p. 
From  these  two  equations  we  obtain 


1/1  +  e  +  1/1  —  e  =  2  cos  J?, 

l/F+7  —  T/f^T  =  2  sin  jp,  (46) 

which  are  convenient  in  many  transformations  of  equations  involving 


e  or 


Equation  (42)  gives 

.     ^      r  sin  v  j9  sin  v 

~~   = 


but  p  =  a  cos2  ^>,  and  6  =  a  cos  ^,  hence 

r  sin  v         cos  ^  sin  v 


.     _, 
sin  E  = 


• 
1  -f-  e  GOBV 

Equation  (41)  gives 

r  cos  v  +  ae  p  cos  v 


or 


_,      ^)  cos  v  +  ae      ag2  cos  v 


and,  putting  a  cos2  (p  instead  of  _p,  and  sin  <p  for  e,  we  get 

cosv-f-e 

cos  E  =  =—  :  -  !  -- 
1  -|-  e  cos  v 

If  we   multiply  the  first  of  equations  (43)  by  cos  \E,  and   the 


56  THEORETICAL   ASTRONOMY. 

second  by  sin^E,  successively  add  and  subtract  the  products,  and 
reduce  by  means  of  the  preceding  equations,  we  obtain 

sin  j  (v  -f-  E)  —  •*/-  cos  \<p  sin  E, 

sin  %(v  —  E)  =  J®  sin  \<p  sin  jEl  (49) 


The  perihelion  distance,  in  an  elliptic  orbit,  is  given  by  the  equa- 
tion 

5  =  o(l—  e). 

21.  The  difference  between  the  true  and  the  mean  anomaly,  or 
v  —  Mj  is  called  the  equation  of  the  centre,  and  is  positive  from  the 
perihelion  to  the  aphelion,  and  negative  from  the  aphelion  to  the 
perihelion.  When  the  body  is  in  either  apsis,  the  equation  of  the 
centre  will  be  equal  to  zero. 

We  have,  from  equation  (39), 


Expanding  this  by  Lagrange's  theorem,  we  get 


+  - 

Let  us  now  take,  equation  (40), 


and,  consequently, 

F(M)  =  (I  —  e  cos  -M")~a. 

Therefore  we  shall  have 

^  =  (1  —  e  cos  M  )~2—  2e2  sin2  M(l  —  e  cos  Jiff* 

d    /  -»\ 

—  e8  -yv>  (sin8  Jf(l  —  e  cos  IT)    )  — 

dM  v 

Expanding  these  terms,  and  performing  the  operations  indicated,  we 

get 

^  =  1  +  2e  cos  M  +  -  (6  xjos2  M  —  4  sin2  Jf) 
-f  |  (16  cos8  M  —  36  sin2  Jf  cos  Jf)  -f- . . . , 


PLACE   IN   THE   ORBIT.  57 

which  reduces  to 


,    (51) 

i 

Equation  (22)  gives 

.        2fdt 
= 


and,  since  /=  }&|/p(l  +  m),  we  have 

dv  =  — ^j-  —  dt,  (52) 

or  ___ 

,  &V 1   -f-  Wl    Cl2      / 

But g =  fa  and  therefore 

a? 

2  2         CX 


By  expanding  the  factor  j/1  —  e2,  we  obtain 
and  hence 


Substituting  for  —  its  value  from  equation  (51),  and  integrating,  we 
get,  since  v  =  0  when  M=0, 

v—M=2e  sin  M  +je2  sin2Jf  +  -^-  (13  sin  33f  —  3  sin  Jf )  +. . .     (53) 

which  is  the  expression  for  the  equation  of  the  centre  to  terms  involving 
e3.  In  the  same  manner,  this  series  may  be  extended  to  higher  powers 
of  e. 

When  the  eccentricity  is  very  small,  this  series  converges  very 
rapidly ;  and  the  value  of  v  —  M  for  any  planet  may  be  arranged  in 
a  table  with  the  argument  M. 

For  the  purpose,  however,  of  computing  the  places  of  a  heavenly 
body  from  the  elements  of  its  orbit,  it  is  preferable  to  solve  the 
equations  which  give  v  and  E  directly ;  and  when  the  eccentricity  is 


58  THEOEETICAL   ASTRONOMY. 

very  great,  this  mode  is  indispensable,  since  the  series  will  not  in 
that  case  be  sufficiently  convergent. 

It  will  be  observed  that  the  formula  which  must  be  used  in  obtain- 
ing the  eccentric  anomaly  from  the  mean  anomaly  is  transcendental, 
and  hence  it  can  only  be  solved  either  by  series  or  by  trial.  But 
fortunately,  indeed,  it  so  happens  that  the  circumstances  of  the  celes- 
tial motions  render  these  approximations  very  rapid,  the  orbits  being 
usually  either  nearly  circular,  or  else  very  eccentric. 

If,  in  equation  (50),  we  put  F(E]  =  E,  and  consequently  F(M] 
=  M,  we  shall  have,  performing  the  operations  indicated  and  reducing, 

E  =  M  -f  e  sin  M  +  ^  sin  2  M  +  &c.  (54) 

Let  us  now  denote  the  approximate  value  of  E  computed  from  this 
equation  by  Eot  then  will 


in  which  &EQ  is  the  correction  to  be  applied  to  the  assumed  value  of  E. 
Substituting  this  in  equation  (39),  we  get 

M=  E0  -f  &EQ  —  e  sin  EQ  —  e  cos  E0*E0; 

and,  denoting  by  MQ  the  value  of  M  corresponding  to  Ew  we  shall 
also  have 

MQ  =  EQ  —  esmE0. 

Subtracting  this  equation  from  the  preceding  one,  we  obtain 

M-M, 

-  --  =  =  A  A.. 

1  —  e  cos  -c/0 

It  remains,  therefore,  only  to  add  the  value  of  &EQ  found  from  this 
formula  to  the  first  assumed  value  of  E,  or  to  Ew  and  then,  using 
this  for  a  new  value  of  E0,  to  proceed  in  precisely  the  same  manner 
for  a  second  approximation,  and  so  on,  until  the  correct  value  of  E  is 
obtained.  When  the  values  of  E  for  a  succession  of  dates,  at  equal 
intervals,  are  to  be  computed,  the  assumed  values  of  E0  may  be  ob- 
tained so  closely  by  interpolation  that  the  first  approximation,  in  the 
manner  just  explained,  will  give  the  correct  value;  and  in  nearly 
every  case  two  or  three  approximations  in  this  manner  will  suffice. 

Having  thus  obtained  the  value  of  E  corresponding  to  M  for  any 
instant  of  time,  we  may  readily  deduce  from  it,  by  the  formulse 
already  investigated,  the  corresponding  values  of  r  and  v. 

In  the  case  of  an  ellipse  of  very  great  eccentricity,  corresponding 
to  the  orbits  of  many  of  the  comets,  the  most  convenient  method  of 


PLACE   IN   THE   CEBIT.  59 

computing  r  and  v,  for  any  instant,  is  somewhat  different.  The 
manner  of  proceeding  in  the  computation  in  such  cases  we  shall  con- 
sider hereafter;  and  we  will  now  proceed  to  investigate  the  formulae 
for  determining  r  and  v,  when  the  orbit  is  a  parabola,  the  formulse 
for  elliptic  motion  not  being  applicable,  since,  in  the  parabola,  a  =  <x> , 
and  e  —  1. 

22.  Observation  shows  that  the  masses  of  the  comets  are  insensible 
in  comparison  with  that  of  the  sun;  and,  consequently,  in  this  case, 
m  —  0  and  equation  (52),  putting  for  p  its  value  2g,  becomes 

k V2q  dt  =  r*dv, 
or 


which  may  be  written 

kdt 
— 3-^  =  2  (1  -f-  tan2  J-y)  sec2  l>vdv  =  (!-}-  tan2  Jv)  d  tan  £0. 

1/2  g1 

Integrating  this  expression  between  the  limits  T  and  t,  we  obtain 

— ^—  =  tan  ^v  +  I  tan8  -^  v,  (55) 

1/2  gf 

which  is  the  expression  for  the  relation  between  the  true  anomaly 
and  the  time  from  the  perihelion,  in  a  parabolic  orbit. 

Let  us  now  represent  by  r0  the  time  of  describing  the  arc  of  a 
parabola  corresponding  to  v  =  90° ;  then  we  shall  have 


2 
or 


T/2-1"3' 


3k  __  4  f 

~ 


Now,  —  ^  is  constant,  and  its  logarithm  is  8.5621876983;  and  if  we 

take  q  =  l,  which  is  equivalent  to  supposing  the  comet  to  move  in 
a  parabola  whose  perihelion  distance  is  equal  to  the  semi-transverse 
axis  of  the  earth's  orbit,  we  find 

log  r0  days  =  2.03987229,  or  r0  =  109.61558  days  ; 
that  is,  a  comet   moving  in  a  parabola  whose   perihelion   distance 


60  THEORETICAL   ASTRONOMY. 

is  equal  to  the  mean  distance  of  the  earth  from  the  sun,  requires 
109.61558  days  to  describe  an  arc  corresponding  to  v  =  90°. 

Equation  (55)  contains  only  such  quantities  as  are  comparable  with 
each  other,  and  by  it  t  —  T,  the  time  from  the  perihelion,  may  be 
readily  found  when  the  remaining  terms  are  known;  but,  in  order 
to  find  v  from  this  formula,  it  will  be  necessary  to  solve  the  equation 
of  the  third  degree,  tan  \v  being  the  unknown  quantity.  If  we  put 
x  =  tan  Ji?,  this  equation  becomes 

ar»  +  3z  —  a  =  0, 

in  which  a  is  the  known  quantity,  and  is  negative  before,  and  positive 
after,  the  perihelion  passage.  According  to  the  general  principle  in 
the  theory  of  equations  that  in  every  equation,  whether  complete  or 
incomplete,  the  number  of  positive  roots  cannot  exceed  the  number 
of  variations  of  sign,  and  that  the  number  of  negative  roots  cannot 
exceed  the  number  of  variations  of  sign,  when  the  signs  of  the  terms 
containing  the  odd  powers  of  the  unknown  quantity  are  changed,  it 
follows  that  when  a  is  positive,  there  is  one  positive  root  and  no 
negative  root.  When  a  is  negative,  there  is  one  negative  root  and 
no  positive  root;  and  hence  we  conclude  that  equation  (55)  can  have 
but  one  real  root. 

We  may  dispense  with  the  direct  solution  of  this  equation  by 
forming  a  table  of  the  values  of  v  corresponding  to  those  of  t  —  T 
in  a  parabola  whose  perihelion  distance  is  equal  to  the  mean  distance 
of  the  earth  from  the  sun.  This  table  will  give  the  time  correspond- 
ing to  the  anomaly  v  in  any  parabola,  whose  perihelion  distance  is 
q,  by  multiplying  by  q2,  the  time  which  corresponds  to  the  same 
anomaly  in  the  table.  We  shall  have  the  anomaly  v  corresponding 
to  the  time  t  —  T  by  dividing  t  —  T  by  (fy  and  seeking  in  the  table 
the  anomaly  corresponding  to  the  time  resulting  from  this  division. 

A  more  convenient  method,  however,  of  finding  the  true  anomaly 
from  the  time,  and  the  reverse,  is  to  use  a  table  of  the  form  gene- 
rally known  as  Barker's  Table.  The  following  will  explain  its  con- 
struction :  — 

Multiplying  equation  (55)  by  75,  we  obtain 

(t  —  T)  =  75  tan  Jv  +  25  tan8  >. 
M=75  tan  ±v  -f  25  tan*  |v, 


1/2  < 
Let  us  now  put 


PLACE   IN   THE   ORBIT.  61 

75k 

and  C0  =  -—  =r,  which  is  a  constant  quantity  ;  then  will 
v  2 


The  value  of  C0  is 

log  <70  =  9.9601277069. 
Again.  let  us  take 

-=?•• 

which  is  called  the  mean  daily  motion  in  the  parabola;  then  will 

M  =  m  (t  —  T)  ==  75  tan  ±v  +  25 


If  we  now  compute  the  values  of  M  corresponding  to  successive 
values  of  v  from  v  =  0°  to  v  =  180°,  and  arrange  them  in  a  table 
with  the  argument  v,  we  may  derive  at  once,  from  this  table,  for  the 
time  (t  —  T)  either  M  when  v  is  known,  or  v  when  M—  m  (t  —  T) 
is  known.  It  may  also  be  observed  that  when  t  —  T  is  negative,  the 
value  of  v  is  considered  as  being  negative,  and  hence  it  is  not  neces- 
sary to  pay  any  further  attention  to  the  algebraic  sign  of.t  —  T  than 
to  give  the  same  sign  to  the  value  of  v  obtained  from  the  table. 

Table  VI.  gives  the  values  of  If  for  values  of  v  from  0°  to  180°, 
with  differences  for  interpolation,  the  application  of  which  will  be 
easily  understood. 

23.  When  v  approaches  near  to  180°,  this  table  will  be  extremely 
inconvenient,  since,  in  this  case,  the  differences  between  the  values  of 
M  for  a  difference  of  one  minute  in  the  value  of  v  increase  very 
rapidly  ;  and  it  will  be  very  troublesome  to  obtain  the  value  of  v 
from  the  table  with  the  requisite  degree  of  accuracy.  To  obviate 
the  necessity  of  extending  this  table,  we  proceed  in  the  following 
manner:  — 

Equation  (55)  may  be  written 


1/2  q 


(1+3  cot2  £ 


and,  multiplying  and  dividing  the  second  member  by  (1  +  cot2  Jt>)*, 
we  shall  ha\e 

F)  ,.          .,,  ' 

_      =  i  tan-  4,  (1  +  cot' 


62  THEORETICAL   ASTRONOMY. 

2 

But  1  +  cot2  Jt;  =  —  -  -  —  —   and  consequently 
sin  v  tan   v 


1/2  5*      ~3sin8v     (1  +  cot2^)3' 

Now,  when  t?  approaches  near  to  180°,  cot^v  will  be  very  small,  and 
the  second  factor  of  the  second  member  of  this  equation  will  nearly 
=  1.  Let  us  therefore  denote  by  w  the  value  of  v  on  the  supposition 
that  this  factor  is  equal  to  unity,  which  will  be  strictly  true  when 
v  =  180°,  and  we  shall  have,  for  the  correct  value  of  v,  the  following 
equation  : 

V  =  W  +  A0, 

AO  being  a  very  small  quantity.     We  shall  therefore  have 


and,  putting  tan  %w  =  0,  and  tan  \  AO  =  x,  we  get,  from  this  equation, 


""" 


6s  1  —  6x  """  (1  —  dx}y 

Multiplying  this  through  by  #3  (1  —  0#)3,  expanding  and  reducing, 
there  results  the  following  equation  : 


+  0*  (2  +  W  +  W  + 
Dividing  through  by  the  coefficient  of  x,  we  obtain 

1  +  30»  _     2  .   02  (2  +  60*  + 

04      1      /5fiNv  "U/     —   •       o   /-«       i       A  aZ      T 


Let  us  now  put 

1  +  30* 


then,  substituting  this  in  the  preceding  equation,  inverting  the  series 
and  reducing,  we  obtain  finally 


But  tan  ^AO  =  x,  therefore 


PLACE   IN   THE   ORBIT.  63 

Substituting  in  this  the  value  of  x  above  found,  and  reducing,  we 
obtain 

—  2  -f  320*  +  1608  +  1008  ,   , 

&C' 


For  all  the  cases  in  which  this  equation  is  to  be  applied,  the  third 
term  of  the  second  member  will  be  insensible,  and  we  shall  have,  to 
a  sufficient  degree  of  approximation, 


Table  VII.  gives  the  values  of  AO,  expressed  in  seconds  of  arc, 
corresponding  to  consecutive  values  of  w  from  w  =  155°  to  w  =  180° 
In  the  application  of  this  table,  we  have  only  to  compute  the  vaiue 
of  M  precisely  as  for  the  case  in  which  Table  VI.  is  to  be  used, 
namely, 

M=m(t  —  T}; 

then  will  w  be  given  by  the  formula 

»[206 

-v-jT 

since  we  have  already  found 


or 


Having  computed  the  value  of  w  from  this  equation,  Table  VII. 
will  furnish  the  corresponding  value  of  A0;  and  then  we  shall  have, 
for  the  correct  value  of  the  true  anomaly, 

v  —  w  -f-  AO, 

which  will  be  precisely  the  same  as  that  obtained  directly  from  Table 
VI.,  when  the  second  and  higher  orders  of  differences  are  taken  into 
account. 

If  v  is  given  and  the  time  t  —  T  is  required,  the  table  will  give, 
by  inspection,  an  approximate  value  of  A0,  using  v  as  argument,  and 
then  w  is  given  by 

w  =  v  —  A. 


64  THEORETICAL   ASTRONOMY. 

The  exact  value  of  A0  is  then  found  from  the  table,  and  hence  we 
derive  that  of  w  •  and  finally  t  —  T  from 

t-T-™    J? 

V  J-      -  yy          *  •         «  • 

GO    Bin*  to 

24.  The  problem  of  finding  the  time  t  —  T  when  the  true  anomaly 
is  given,  may  also  be  solved  conveniently,  and  especially  so  when  v  is 
small,  by  the  following  process:  — 

Equation  (55)  is  easily  transformed  into 


^-= 

3v 


from  which  we  obtain,  since  q  =  r  cos2|v, 

~2^~          \  1/2  /  ~      \  1/2  /  * 

Let  us  now  put 

sin^v 

sm  x  =  — 7==, 
1/2 
and  we  have 

— - — - — -  =  3  sin  x  —  4  sin's  =  sin  3#. 
2r* 

Consequently, 


which  admits  of  an  accurate  and  convenient  numerical  solution.     To 
facilitate  the  calculation  we  put 

~-  _  sin  3x 

J\  — 


sin  v 


the  values  of  which  may  be  tabulated  with  the  argument  v.  "When 
v  —  0,  we  shall  have  N=  fl/2,  and  when  v  =  90,  we  have  N=  1  ; 
from  which  it  appears  that  the  value  of  ^V  changes  slowly  for  values 
of  v  from  0°  to  90°.  But  when  v  =  180°,  we  shall  have  N=  * 
and  hence,  when  v  exceeds  90°,  it  becomes  necessary  to  introduce  au 
auxiliarv  different  from  N.  We  shall,  therefore,  put  in  this  case, 

W  =  JV  sin  v  =  sin  3#; 


PLACE   IN   THE   ORBIT.  65 

from  which  it  appears  that  Nf=  1  when  v  =  90°,  and  that  JV'  =  Jv'iB 
when  v  =  180°.     Therefore  we  have,  finally,  when  v  is  less  than  90°, 

2        f 
t—  T=-7c1-Nr  sinv, 

OK 

and,  when  v  is  greater  than  90°, 


o 
in  which  log  —  =  1.5883272995,  from  which  t  —  T  is  easily  derived 

oA/ 
when  v  is  known. 

Table  VIII.  gives  the  values  of  N,  with  diiferences  for  interpola- 
tion, for  values  of  v  from  v  =  0°  to  v  =  90°,  and  the  values  of  Nr 
for  those  of  v  from  v  =  90°  to  v  =  180°. 

25.  We  shall  now  consider  the  case  of  the  hyperbola,  which  differs 
from  the  ellipse  only  that  e  is  greater  than  1  ;  and,  consequently,  the 
formulae  for  elliptic  and  hyperbolic  motion  will  differ  from  each  other 
only  that  certain  quantities  which  are  positive  in  the  ellipse  are  nega- 
tive or  imaginary  in  the  hyperbola.  We  may,  however,  introduce 
auxiliary  quantities  which  will  serve  to  preserve  the  analogy  between 
the  two,  and  yet  to  mark  the  necessary  distinctions. 

For  this  purpose,  let  us  resume  the  equation 


T  = 


2  cos  J-  (v  +  4)  cos  J  (v  —  4)' 

When  v  —  0,  the  factors  cos  |  fa  +  ^  an^  cos£(v  —  ^)  in  the  de- 
nominator will  be  equal ;  and  since  the  limits  of  the  values  of  v  are 
180°— 4>  and  —(180°  —  $),  it  follows  that  the  first  factor  will  vanish 
for  the  maximum  positive  value  of  v,  and  that  the  second  factor  will 
vanish  for  the  maximum  negative  value  of  v,  and,  therefore,  that,  in 
either  case,  r  =  oo. 

In  the  hyperbola,  the  semi-transverse  axis  is  negative,  and,  conse- 
quenlly,  we  have,  in  this  case, 

p==a(e2  —  1),  or  a  —p  cot* 4. 

We  have,  also,  for  the  perihelion  distance, 

q  =  a(e  —  1). 
Let  us  now  put  

tan  IF  =  tan  '  (56) 


66  THEOKETICAL  ASTRONOMY. 

which  is  analogous  to  the  formula  for  the  eccentric  anomaly  E  in  an 

ellipse:  and.  since  e=  -  ,  we  shall  have 
cos  4* 


_ 

e+l~l  +  COS+~ 

and,  consequently, 

tan  \F  =  tan  ^v  tan  ^.  (57) 

We  shall  now  introduce  an  auxiliary  quantity  <r,  such  that 


whence  we  derive 

t  n        ff  —  1  /Kfcv 

tan  JJF1  =      ,  ]»  (58) 

and  also 

ff==cMl(v  —  *\  (59) 

cos  £  (v  -j-  4") 

This  last  equation  shows  that  cr  =  1  when  the  comet  is  in  its  perihe- 
lion; a=  oo  when  v=180° — ^;  and  0=0  when  v  =  —  (180°  —  ^> 

2tan|^  ,    ,,  , 

Since  tan  F= TTTr'  we  shall  nave 

1  —  tan'  %F 

a  —  l\ 

i  1  \ 

(60) 


-  1^=1] 

Wi/ 


Squaring  this  equation,  adding  1  to  both  members,  and  reducing  we 
obtain 


cos 
Replacing  <r  in  this  equation  by  its  value  from  equation  (59),  we  get 


or 

1       _  1  -f  cos  v  cos  ^  _  (e  +  cos  v)  cos  4> 

cosF  ~~  2cosi(^H-  *)  cos  i  (v  —  4)  "~  2  cos  £  (v  -f-  4)  cos  £  (v  — 


which  reduces  to 

1  ~.  f  ~        I        ^/vc,«A 

(62) 


PLACE   IN   THE   ORBIT.  67 

If  we  add  ip  1  to  both  members  of  this  equation,  we  shall  havp 


coaF  p 

Taking  first  the  upper  sign,  and  then  the  lower  sign,  and  reducing, 
we  get 


VcosF 


1/r  cos  %>  =         _--)  cog  ,  ^  (63) 

VcosF 

These  equations  for  finding  r  and  v,  it  will  be  observed,  are  analogous 
to  those  previously  investigated  for  an  elliptic  orbit.  These  equations 
give,  by  division, 


ten  v  = 


which  is  identical  with  the  equation  (56),  and  may  be  employed  to 
verify  the  computation  of  r  and  v. 

Multiplying  the  last  of  equations  (63)  by  the  first,  putting  for 
£  —  1  its  value  tan2  ij/,  and  reducing,  we  obtain 

r  sin  v  =  a  tan  4/  tan  F=£a  tan  4  1  <r  —  -I.  (64) 

Further,  we  have 

v  cos  v  ar(e  -4-  cos  v) 

r  cos  v  =  r-^—        —  =  ae  --  !  -  , 

1  -f-  6  COS  V  p 

which,  combined  with  equation  (62),  gives 

rcosv  =  a\e  --  r7l  =  2a|2e  —  *  —  -)•  (65) 

\         cos  F  /          \  *J 

if  we  square  these  values  of  r  sinv  and  r  cosv,  add  the  results  to- 
gether, reduce,  and  extract  the  square  root,  we  find 

2)-         (66) 

We  might  also  introduce  the  auxiliary  quantity  a  into  the  equations 
(63);  but  such  a  transformation  is  hardly  necessary,  and,  if  at  all 
desirable,  it  can  be  easily  effected  by  means  of  the  formulae  which  we 
have  already  derived. 


68  THEORETICAL  ASTRONOMY. 

26.  Let  us  now  resume  the  equation 

cos    (v  —  4 


_ 

~~ 


cos  $  0  +  4)' 
Differentiating  this,  regarding  ^  as  constant,  we  have 

,  sin  4  , 

=  2co8*iO  +  4)     ' 

and,  dividing  this  equation  by  the  preceding  one,  we  get 
dff  sin  4 


ff        2  cos  £  (v  +  40  cos  2  (v  —  40 
But 

p  cos  ^ 


__ 
~ 


2  cos  £  (v  H-  4)  cos  £  (v 
consequently, 

d<r  _  r  tan  4   _ 
V  "     "~^~     Vj 
which  gives 


ff  tan  4 


Substituting  this  value  of  i*dv  in  equation  (22),  and  putting  instead 
of  2/  its  value  ~kVpy  from  equation  (30),  the  mass  being  considered  as 
insensible  in  comparison  with  that  of  the  sun,  we  get 


Then,  substituting  for  r  its  value  from  equation  (66),  and  for  p  its 
value  a  tan2  ^/,  we  have 

IcVp  dt  =  a?  tan  *  I  \e  (  1  +  i)  —  1)  dff. 
Integrating  this  between  the  limits  T  and  t,  we  obtain 

*l/5(«—  T)  =  a»tan^Je(*  —  i)—  loge<7J,  (67) 

in  which  loge  <r  is  the  Naperian  or  hyperbolic  logarithm  of  ff.    Sinoe 
Vp  =  Va  tan  ij/,  if  we  put 


PLACE   IN   THE   ORBIT.  69 

in  which  v  is  the  mean  daily  motion;  and  if  we  also  put 


in  which  N0  corresponds  to  the  mean  anomaly  M  in  an  ellipse,  we 
shall  have,  from  equation  (67), 

iog.«.  (68) 

If  we  multiply  both  members  of  this  equation  by  A  =  0.434294482, 
the  modulus  of  the  common  system  of  logarithms,  and  put 


we  shall  have 

JV=  \ek  I  <T  —  -  1  —  log  ff, 

wherein  log  A  =  9.6377843113,  and  logM  =  7.8733657527. 

Let  us  now  introduce  J^into  this  formula;  and  for  this  purpose  we 
have 


and  also 

log  ff  =  log  tan  (45°  -f  JJP). 

Therefore  we  obtain 

N  =  el  tan  F  —  log  tan  (45°  +  $F).  (69) 

This  equation  will  give,  directly,  the  time  t  —  T  from  the  perihelion, 
when  a,  e,  and  .Fare  known;  but,  since  it  is  transcendental,  in  the 
solution  of  the  inverse  problem,  that  of  finding  the  true  anomaly 
and  radius-  vector  from  the  time,  the  value  of  F  can  only  be  found  by 
successive  approximations. 

If  we  differentiate  the  last  equation,  regarding  ^7"  and  F  as  vari- 
able, we  get 


hEence,  if  we  denote  an  approximate  value  of  F  by  Fn  and  the  cor- 
responding value  of  N  by  Nn  the  correction  *F,  to  the  assumed  value 
of  F  may  be  computed  by  the  formula 


F_- 
'     '    Ke  —  cosF,} 


70  THEORETICAL   ASTRONOMY. 

This  correction  being  applied  to  Fn  a  nearer  approximation  to  the 
true  value  of  F  will  be  obtained;  and  by  repeating  the  operation 
there  results  a  still  closer  approximation.  This  process  may  be  con- 
tinued until  the  exact  value  of  F  is  found,  and,  when  several  suc- 
cessive places  are  required,  the  first  assumed  value  may  be  estimated, 
in  advance,  so  closely  that  a  very  few  trials  will  suffice.  In  practice, 
however,  cases  will  rarely  occur  in  which  this  formula  will  be  applied, 
since  the  probability  of  hyperbolic  motion  is  small,  and,  whenever 
any  positive  indication  of  an  eccentricity  greater  than  1  has  been 
found  to  exist,  it  has  only  been  after  a  very  accurate  series  of  observa- 
tions has  been  introduced  as  the  basis  of  the  calculation.  For  a 
majority  of  the  cases  which  do  really  occur,  the  most  accurate  and 
convenient  method  of  finding  r  and  v  will  be  explained  hereafter. 

27.  If  we  consider  the  equation 

M=E  —  esmE, 

we  shall  see  that,  when  logarithms  of  six  or  seven  decimals  are  used, 
the  error  which  may  exist  in  the  determination  of  E  when  M  and  e 
are  given,  will  increase  as  e  increases,  but  in  a  much  greater  ratio; 
and,  when  the  eccentricity  becomes  nearly  equal  to  that  of  the  para- 
bola, the  error  may  be  very  great.  In  the  case  of  hyperbolic  motion, 
also,  the  numerical  solution  of  equation  (69),  when  e  —  1  is  very 
small,  and  with  the  ordinary  logarithmic  tables,  becomes  very  un- 
certain. This  can  only  be  remedied,  when  equations  (39)  and  (69) 
are  employed,  by  using  more  extended  logarithmic  tables;  and  when 
the  orbit  diifers  only  in  an  extremely  slight  degree  from  a  parabola, 
even  with  the  most  extended  logarithmic  tables  which  have  been 
constructed,  the  error  may  be  very  large.  For  this  reason  we  have 
recourse  to  other  methods,  which  will  give  the  required  accuracy 
without  introducing  inconveniences  which  are  proportionally  great. 

We  shall,  therefore,  now  proceed  to  develop  the  formulae  for  find- 
ing the  true  anomaly  in  ellipses  and  hyperbolas  which  differ  but 
little  from  the  parabola,  such  that  they  will  furnish  the  required 
accuracy,  when  the  exact  solution  of  equations  (39)  or  (69)  with  the 
logarithmic  tables  in  common  use  is  impossible. 

For  this  purpose,  let  us  resume  equation  (22),  which,  by  substi- 
tuting for  2/  its  value  &I/JP,  the  mass  of  the  comet  being  neglected 
in  comparison  with  that  of  the  sun,  becomes 

1ci/pdt'=r'ldv, 


PLACE  IN  THE  ORBIT.  71 

or 

dt  = 


(1  -f-  e  cos  v 

Let  us  now  put  u  =  tanjv,  and  we  shall  have 

1—  u*  2du 


COSV  — 


J.  W  -•  +l\Ai\AI 

=ru*'  '  ~ru 


Substituting  these  values  in  the  preceding   equation,  and  putting 
1—  e  _  . 

_  _  2p*    (1  -f  w2)  du 
or,  since  p  =  q  (1  +  e), 


Let  us  now  develop  the  second  member  into  a  series.     This  may  be 
written  thus: 


and  developing  the  last  factor  into  a  series,  we  obtain 

(1  +  m2)-2  =  1  —  2m'  +  3iV  —  4iV  +  &c. 
Consequently, 

m«)-3  =  i  4.  tt«  _  2»(t*«  +  w4)  + 


Multiplying  this  equation  through  by  du,  and  integrating  between 
the  limits  T  and  t,  the  result  is 


—  Aff  (>'  +  X)  +  &c- 

In  the  case  of  the  parabola,  e  =  1  and  i  —  0,  and  this  equation  becomes 
identical  with  (55). 
Let  us  now  put 


72  THEORETICAL   ASTRONOMY. 

and  also 


then  the  angle  V  will  not  be  the  true  anomaly  in  the  parabola,  but 
an  angle  derived  from  the  solution  of  a  cubic  equation  of  the  same 
form  as  that  for  finding  the  parabolic  anomaly;  and  its  value  may 
be  found  by  means  of  Table  VI.,  if  we  use  for  M  the  value  com- 
puted from 

75K^—  _T)      \T±2 

-      3  *    \          9        • 

1/2  f 
Let  U  be  expanded  into  a  series  of  the  form 


which  is  evidently  admissible,  a,  /?,  fy  ____  being  functions  of  u  and 
independent  of  i.  It  remains  now  to  determine  the  values  of  the 
coefficients  a,  /?,  f,  &c.,  and,  in  doing  so,  it  will  only  be  necessary  to 
consider  terms  of  the  third  order,  or  those  involving  i3,  since,  for 
nearly  all  of  those  cases  in  which  the  eccentricity  is  such  that  terms 
of  the  order  i4  will  sensibly  affect  the  result,  the  general  formulae 
already  derived,  with  the  ordinary  means  of  solution,  will  give  the 
required  accuracy.  We  shall,  therefore,  have 

U  +  I  U3  =  U  4-  ai  +  &  4  rP  4  l(u  4  ai  4-  &  +  rW, 
or,  again  neglecting  terms  of  the  order  i*, 


But  we  have  already  found,  (70), 

k(t—T)i/l  +  e  =  n  +    ^  =  u  +        _  2.     u,  +       ^ 

2gf 

+  3?  (iw5  +  >7)  —  4i3  (>7  +  >9). 

Since  the  first  members  of  these  equations  are  identical,  it  follows,  by 
the  principle  of  indeterminate  coefficients,  that  the  coefficients  of  the 
like  powers  of  i  are  equal,  and  we  shall,  therefore,  have 


ua*  +  (i  _|_  w.)  p  =  4.  3  (><  -f  4rT), 

3  4-  2wa/9  4-  (1  +  w2)  r  =  —  4  (>7  4-  ^U9). 


From  the  first  of  these  equations  we  find 


PLACE   IN   THE   CEBIT.  73 


The  second  equation  gives 

3(K  +  ju*)-n.« 
l  +  u« 

or,  substituting  for  a  its  value  just  found,  and  reducing, 

g  =  3  0*  +  f-H"7  +  jft  *  +  jftu") 

(1  +  **)• 

We  have  also 

_  —  4  (i^7  +  X)  —  j»3  —  2a^ti 
r"  1  +  u2 

and  hence,  substituting  the  values  of  oc  and  ft  already  found,  and 
reducing,  we  obtain  finally 


(I  -I-  < 

Again,  we  have 

tan     U —  tan     (u  — {—  oil 

Developing  this,  and  neglecting  terms  of  the  order  i4,  we  get 
tan^U^f 


Now,  since  w  =  tan  Ji?  and  U=  tan  \  F,  we  shall  have 


or 

2a     . 


,,  (72) 


Substituting  in  this  equation  the  values  of  a,  ft,  and  ?  already  found, 
and  reducing,  we  obtain  finally 


(!+«')' 


THEORETICAL   ASTRONOMY. 

This  equation  can  be  used  whenever  the  true  anomaly  in  the 
ellipse  or  hyperbola  is  given,  and  the  time  from  the  perihelion  is  to 
be  determined.  Having  found  the  value  of  F,  we  enter  Table  VI. 
with  the  argument  F  and  take  out  the  corresponding  value  of  M; 
and  l,hen  we  derive  t  —  T  from 

t_T=M< 


in  which  log  C0  =  9.96012771. 

For  the  converse  of  this,  in  which  the  time  from  the  perihelion  is 
given  and  the  true  anomaly  is  required,  it  is  necessary  to  express  the 
difference  v  —  F  in  a  series  of  ascending  powers  of  i,  in  which  the 
coefficients  are  functions  of  U.  Let  us,  therefore,  put 

u  =  U  4-  o'i  -f  /3'i2  -f  r'$  +  &c. 

Substituting  this  value  of  u  in  equation  (70),  and  neglecting  terms 
multiplied  by  i4  and  higher  powers  of  i,  we  get 


_  4  [73^2  _  2  Ua'2  —  4  U7  —  1  17')  t". 

But,  since  the  first  member  of  this  equation  is  equal  to  U  +  J  Z7S,  we 
shall  have,  by  the  principle  of  indeterminate  coefficients, 


From  these  equations,  we  find 


s  _  Iff  W  +  ill!  u9  +  VftV  ff"  +  f  If  u13 + fill  0™ + i-VV 

(i  +  tf 2)6 

If  we  interchange  v  and  F  in  equation  (72),  it  becomes,  writing  a;, 
0',  f'  for  a,  /9,  ^, 


_  T7  , 

* 


PLACE   IN   THE   ORBIT.  75 

2/3'  2a'2i7 


+  /_^  ___  4a^^7_      2(t7^j)      \ 
Ml  +  tf2       (1  +  tf2)2      (1  +  tf2)3       / 

Substituting  in  this  equation  the  above  values  of  cc/,  /?',  and  p',  and 
reducing,  we  obtain,  finally, 


_F[  .    , 

22 


.   If  I  ^T+  JIM  ^9+  f  If  f  I  ^  n+  If  If  ^18 

2« 


by  means  of  which  v  may  be  determined,  the  angle  V  being  taken 
from  Table  VI.,  so  as  to  correspond  with  the  value  of  M  derived 
from 


Equations  (73)  and  (74)  are  applicable,  without  any  modification, 
to  the  case  of  a  hyperbolic  orbit  which  differs  but  little  from  the 
parabola.  In  this  case,  however,  e  is  greater  than  unity,  and,  conse- 
quently, i  is  negative. 

28.  In  order  to  render  these  formulae  convenient  in  practice,  tables 
may  be  constructed  in  the  following  manner:  — 
Let  x  =  v  or  F,  and  tan  Ja?  =  6,  and  let  us  put 

<0*+|0s 
100(1  -f-  02)2  ' 

^ 


10000  (1 


ii 

10000  (1  +  02)4 


1000000(1  +  02)6 

+  iiii^9  +  flff^11  +  If  ^13  +  jiff*"  + 


1000000  (1  +  ^2)6 

wherein  s  expresses  the  number  of  seconds  corresponding  to  the 
length  of  arc  equal  to  the  radius  of  a  circle,  or  log  s==  5.31442513. 
We  shall,  therefore,  have  :  — 


v=V+A  (1000  +  5(10017+ 


76  THEORETICAL  ASTRONOMY. 

and,  when  x  =  v, 

V=v-A (lOOi)  +  & (lOOi)2 -  C' (lOOi)8. 

Table  IX.  gives  the  values  of  A,  B,  B',  <7,  and  C'  for  consecu- 
tive values  of  x  from  x  =  0°  to  x  =  149°,  with  differences  for  inter- 
polation. 

When  the  value  of  v  has  been  found,  that  of  r  may  be  derived 
from  the  formula 

r=  g(l-M)  ^ 
1  -j-  e  cos  v 

Similar  expressions  arranged  in  reference  to  the  ascending  powers 
of  (1  —  e)  or  of  I  I  —^  -  1  —  II  may  be  derived,  but  they  do  not  con- 

«2     \^        \ 
I  —  1  I  is 
1  -f~  &  '  i 

than  i,  yet  the  coefficients  are,  in  each  case,  so  much  greater  than 
those  of  the  corresponding  powers  of  i,  that  three  terms  will  not 
afford  the  same  degree  of  accuracy  as  the  same  number  of  terms  in 
the  expressions  involving  i. 

29.  Equations  (73)  and  (74)  will  serve  to  determine  v  or  t  —  T  in 
nearly  all  cases  in  which,  with  the  ordinary  logarithmic  tables,  the 
general  methods  fail.  However,  when  the  orbit  differs  considerably 
from  a  parabola,  and  when  v  is  of  considerable  magnitude,  the  results 
obtained  by  means  of  these  equations  will  not  be  sufficiently  exact, 
and  we  must  employ  other  methods  of  approximation  in  the  case  that 
the  accurate  numerical  solution  of  the  general  formulae  is  still  impos- 
sible. It  may  be  observed  that  when  E  or  F  exceeds  50°  or  60°,  the 
equations  (39)  and  (69)  will  furnish  accurate  results,  even  when  e 
differs  but  little  from  unity.  Still,  a  case  may  occur  in  which  the 
perihelion  distance  is  very  small  and  in  which  v  may  be  very  great 
before  the  disappearance  of  the  comet,  such  that  neither  the  general 
method,  nor  the  special  method  already  given,  will  enable  us  to  de- 
termine v  or  t  —  T  with  accuracy;  and  we  shall,  therefore,  investigate 
another  method,  which  will,  in  all  cases,  be  sufficiently  exact  when 
the  general  formulae  are  inapplicable  directly.  For  this  purpose,  let 
us  resume  the  equation 

=  E —  e  sin  Et 


PLACE   IN   THE   ORBIT.  77 

which,  since  q  =  a(l  —  e),  may  be  written 


If  we  put 


we  shall  have 


^e       201/1  \      1     l  +  9e      » 

"  * 


2 

Let  us  now  put 


9E  -f  sin  .E 


201/2 

and 


then  we  have 


When  jB  is  known,  the  value  of  w  may,  according  to  this  equation, 
be  derived  directly  from  Table  VI.  with  the  argument 


and  then  from  w  we  may  find  the  value  of  A.  It  remains,  therefore, 
to  find  the  value  of  B;  and  then  that  of  v  from  the  resulting  value 
of  A. 

Now,  we  have 


2  tan 


and  if  we  put  tan2  \E  =  r,  we  get 


Sn       =  ==    *         -^       r 

We  have,  also, 

E  =  2  tan"1  r^=  2r^  (1  —  JT  +  Jt»  -  ^r8  +  Ac.)- 


78  THEOKETICAL  ASTRONOMY. 

Therefore, 

15  (E  —  sin  E)  =  2t*  (10r  — 


and 

9E  +  sin  E  =  2r^  (10  —  ^  T  +  y  r»      y  r8  +  if  i*  —  &c.). 

Hence,  by  division, 


and,  inverting  this  series,  we  get 


which  converges  rapidly,  and  from  which  the  value  of  —  may  be 
found. 

Let  us  now  put 

A_    l_ 

r~02' 

then  the  values  of  O  may  be  tabulated  with  the  argument  A;  and, 
besides,  it  is  evident  that  as  long  as  A  is  small  C2  will  not  differ 
much  from  1  -f-  \A. 

Next,  to  find  B,  we  have 


-  40.), 

and  hence 

_  62    .        9 

~~~  ""     T^      ~  2525^+  33eST6T    — 


from  which  we  easily  find 

S  =  1  +  T|s^2  +  ,|^s  +  riW,^4  +  &c. 
If  we  compare  equations  (44)  and  (56),  we  get 


tan  ±E  =  T  tan  J.P. 

Hence,  in  the  case  of  a  hyperbolic  orbit,  if  we  put  tan2|.F—  r',  we 
must  write  —  r'  in  place  of  r  in  the  formulae  already  derived  ;  and, 
from  the  series  which  gives  A  in  terms  of  r,  it  appears  that  A  is  in 
this  case  negative.  Therefore,  if  we  distinguish  the  equations  for 


PLACE   IN   THE   CEBIT.  79 

Hyperbolic  motion  from  those  for  elliptic  motion  by  writing  A',  Bft 
and  Cf  in  place  of  A,  B,  and  C,  respectively,  we  shall  have 


"  -  Ac. 

Table  X.  contains  the  values  of  log  B  and  log  C  for  the  ellipse 
and  the  hyperbola,  with  the  argument  A,  from  A  =  0  to  A  =  0.3, 
For  every  case  in  which  A  exceeds  0.3,  the  general  formulae  (39) 
and  (69)  may  be  conveniently  applied,  as  already  stated. 

The  equation 

Hi  Q 

tan  |v  =  -v/  T—5--  tan  \E 
gives 


JL.  & 

or,  substituting  the  value  of  A  in  terms  of  w, 

tan^=(7tan>^^l±^.  (76) 

The  last  of  equations  (43)  gives 


~l+tanfj-tf 
Hence  we  derive 


The  equation  for  v  in  a  hyperbolic  orbit  is  of  precisely  the  same  form 
as  (76),  the  accents  being  omitted,  and  the  value  of  A  being  computed 
from 


For  the  radius-vector  in  a  hyperbolic  orbit,  we  find,  by  means  of  the 
last  of  equations  (63), 

T==(l— 


When  t  —  T  is  given  and  r  and  v  are  required,  we  first  assume 
B  =  1  ,  and  enter  Table  VI.  with  the  argument 


i1         B 


80  THEORETICAL   ASTRONOMY. 

in  which  log  CQ  =  9.96012771,  and  take  out  the  corresponding  value 
of  w.     Then  we  derive  A  from  the  equation 


in  the  case  of  the  ellipse,  and  from  (78)  in  the  case  of  a  hyperbolic 
orbit.  With  the  resulting  value  of  A,  we  find  from  Table  X.  the 
corresponding  value  of  log  JB,  and  then,  using  this  in  the  expression 
for  My  we  repeat  the  operation.  The  second  result  for  A  will  not 
require  any  further  correction,  since  the  error  of  the  first  assumption 
of  B  =  1  is  very  small  ;  and,  with  this  as  argument,  we  derive  the 
value  of  log  C  from  the  table,  and  then  v  and  r  by  means  of  the 
equations  (76)  and  (77)  or  (79). 

When  the  true  anomaly  is  given,  and  the  time  t  —  T  is  required, 
we  first  compute  T  from 


in  the  case  of  the  ellipse,  or  from 


in  the  case  of  the  hyperbola.  Then,  with  the  value  of  r  as  argu- 
ment, we  enter  the  second  part  of  Table  X.  and  take  out  an  approxi- 
mate value  of  A,  and,  with  this  as  argument,  we  find  logJ?  and  log  C. 
The  equation 


will  show  whether  the  approximate  value  of  A  used  in  finding 
log  C  is  sufficiently  exact,  and,  hence,  whether  the  latter  requires  any 
correction.  Next,  to  find  w,  we  have 


5(1  -H)' 

and,  with  w  as  argument,  we  derive  M  from  Table  VI.     Finally,  we 
have 

(80) 


by  means  of  which  the  time  from  the  perihelion  may  be  accurately 
determined. 


POSITION   IN   SPACE.  81 

30.  We  have  thus  far  treated  of  the  motion  of  the  heavenly  bodies, 
relative  to  the  sun,  without  considering  the  positions  of  their  orbits 
in  space ;  and  the  elements  which  we  have  employed  are  the  eccen- 
tricity and  semi-transverse  axis  of  the  orbit,  and  the  mean  anomalv 
at  a  given  epoch,  or,  what  is  equivalent,  the  time  of  passing  thft 
perihelion.  These  are  the  elements  which  determine  the  position  of 
the  body  in  its  orbit  at  any  given  time.  It  remains  now  to  fix  its 
position  in  space  in  reference  to  some  other  point  in  space  from  which 
we  conceive  it  to  be  seen.  To  accomplish  this,  the  position  of  its 
orbit  in  reference  to  a  known  plane  must  be  given ;  and  the  elements 
which  determine  this  position  are  the  longitude  of  the  perihelion,  the 
longitude  of  the  ascending  node,  and  the  inclination  of  the  plane  of 
the  orbit  to  the  known  plane,  for  which  the  plane  of  the  ecliptic  is 
usually  taken.  These  three  elements  will  enable  us  to  determine  the 
co-ordinates  of  the  body  in  space,  when  its  position  in  its  orbit  has 
been  found  by  means  of  the  formula  already  investigated. 

The  longitude  of  the  ascending  node,  or  longitude  of  the  point 
through  which  the  body  passes  from  the  south  to  the  north  side  of 
the  ecliptic,  which  we  will  denote  by  & ,  is  the  angular  distance  of 
this  point  from  the  vernal  equinox.  The  line  of  intersection  of  the 
plane  of  the  orbit  with  the  fundamental  plane  is  called  the  line  of 
nodes. 

The  angle  which  the  plane  of  the  orbit  makes  with  the  plane  of 
the  ecliptic,  which  we  will  denote  by  i,  is  called  the  inclination  of 
the  orbit.  It  will  readily  be  seen  that,  if  we  suppose  the  plane  of 
the  orbit  to  revolve  about  the  line  of  nodes,  when  the  angle  i  exceeds 
180°,  &  will  no  longer  be  the  longitude  of  the  ascending  node,  but 
will  become  the  longitude  of  the  descending  node,  or  of  the  point 
through  which  the  planet  passes  from  the  north  to  the  south  side  of 
the  ecliptic,  which  is  denoted  by  t5,  and  which  is  measured,  as  in  the 
case  of  &  ?  from  the  vernal  equinox. 

It  will  easily  be  understood  that,  when  seen  from  the  sun,  so  long 
as  the  inclination  of  the  orbit  is  less  than  90°,  the  motion  of  the 
body  will  be  in  the  same  direction  as  that  of  the  earth,  and  it  is  then 
said  to  be  direct.  When  the  inclination  is  90°,  the  motion  will  be  at 
right  angles  to  that  of  the  earth ;  and  when  i  exceeds  90°,  the  motion 
in  longitude  will  be  in  a  direction  opposite  to  that  of  the  earth,  and 
it  is  then  called  retrograde.  It  is  customary,  therefore,  to  extend  the 
inclination  of  the  orbit  only  to  90°,  and  if  this  angle  exceeds  a  right 
angle,  to  regard  its  supplement  as  the  inclination  of  the  orbit,  noting 
simply  the  distinction  that  the  motion  is  retrograde. 


82  THEORETICAL    ASTRONOMY. 

The  I'jngitude  of  the  perihelion,  which  is  denoted  by  x,  fixes  the 
position  of  the  orbit  in  its  own  plane,  and  is,  in  the  case  of  direct 
motion,  the  sum  of  the  longitude  of  the  ascending  node  and  the 
angular  distance,  measured  in  the  direction  of  the  motion,  of  the 
perihelion  from  this  node.  It  is,  therefore,  the  angular  distance  of 
the  perihelion  from  a  point  in  the  orbit  whose  angular  distance  back 
from  the  ascending  node  is  equal  to  the  longitude  of  this  node;  or 
it  may  be  measured  on  the  ecliptic  from  the  vernal  equinox  to  the 
ascending  node,  then  on  the  plane  of  the  orbit  from  the  node  to  the 
place  of  the  perihelion. 

In  the  case  of  retrograde  motion,  the  longitudes  of  the  successive 
points  in  the  orbit,  in  the  direction  of  the  motion,  decrease,  and  the 
point  in  the  orbit  from  which  these  longitudes  in  the  orbit  are 
measured  is  taken  at  an  angular  distance  from  the  ascending  node 
equal  to  the  longitude  of  that  node,  but  taken,  from  the  node,  in  the 
same  direction  as  the  motion.  Hence,  in  this  case,  the  longitude  of 
the  perihelion  is  equal  to  the  longitude  of  the  ascending  node  dimi- 
nished by  the  angular  distance  of  the  perihelion  from  this  node. 

It  may,  perhaps,  seem  desirable  that  the  distinctions,  direct  and 
retrograde  motion,  should  be  abandoned,  and  that  the  inclination  of 
the  orbit  should  be  measured  from  0°  to  180°,  since  in  this  case 
one  set  of  formulae  would  be  sufficient,  while  in  the  common  form 
two  sets  are  in  part  required.  However,  the  custom  of  astronomers 
seems  to  have  sanctioned  these  distinctions,  and  they  may  be  per- 
petuated or  not,  as  may  seem  advantageous. 

Further,  we  may  remark  that  in  the  case  of  direct  motion  the  sum 
of  the  true  anomaly  and  longitude  of  the  perihelion  is  called  the 
true  longitude  in  the  orbit;  and  that  the  sum  of  the  mean  anomaly 
and  longitude  of  the  perihelion  is  called  the  mean  longitude,  an  ex- 
pression which  can  occur  only  in  the  case  of  elliptic  orbits. 

In  the  case  of  retrograde  motion  the  longitude  in  the  orbit  is  equal 
to  the  longitude  of  the  perihelion  minus  the  true  anomaly. 

31.  We  will  now  proceed  to  derive  the  formulae  for  determining 
the  co-ordinates  of  a  heavenly  body  in  space,  when  its  position  in  its 
orbit  is  known. 

For  the  co-ordinates  of  the  position  of  the  body  at  the  time  ts  we 
have 

x  =  r  cos  v, 
3  =  r  sin  v, 


POSITION  IN  SPACE.  83 

the  line  of  apsides  being  taken  as  the  axis  of  x,  and  the  origin  being 
taken  at  the  centre  of  the  sun. 

If  we  take  the  line  of  nodes  as  the  axis  of  x,  we  shall  have 

x  =  r  cos  (v  -f  w), 
y  =  r  sin  (v  -f-  w), 

to  being  the  arc  of  the  orbit  intercepted  between  the  place  of  the 
perihelion  and  of  the  node,  or  the  angular  distance  of  the  perihelion 
from  the  node. 

Ndw,  we  have  co  =  7r  —  ft  in  the  case  of  direct  motion,  and  a)  -— 
ft  —  TT  in  the  case  of  retrograde  motion ;  and  hence  the  last  equations 

become 

x  =  r  cos  (v  ±  TT  qp  ft) 

y  =  r  sin  (v  db  ?r  qp  ft) 

the  upper  and  lower  signs  being  taken,  respectively,  according  as 
the  motion  is  direct  or  retrograde.  The  arc  v±7rq=ft=wis  called 
the  argument  of  the  latitude. 

Let  us  now  refer  the  position  of  the  body  to  three  co-ordinate 
planes,  the  origin  being  at  the  centre  of  the  sun,  the  ecliptic  being 
taken  as  the  plane  of  xy,  and  the  axis  of  #,  in  the  line  of  nodes. 

Then  we  shall  have 

x'  =  r  cos  u, 

yf  =  db  r  sin  u  cos  i, 
z'  —  r  sin  u  sin  i. 

If  we  denote  the  heliocentric  latitude  and  longitude  of  the  body,  at 
the  time  t,  by  b  and  I,  respectively,  we  shall  have 

x'  =  r  cos  b  cos  (I  —  ft  ), 
yf  =  r  cos  b  sin  (I  —  ft  ), 

/  =  r  sin  b, 
and,  consequently, 

cos  u  =  cos  b  cos  (I  —  &), 

±  sin  u  cos  i  =  cos  b  sin  (7  —  ft),  (81) 

sin  w  sin  i  =  sin  6. 

From  these  we  derive 

tan  (I  —  ft  )  —  ±  tan  w  cos  i, 

tan  6  =  db  tan  i  sin  (f  —  ft  ),  (82) 

which  serve  to  determine  I  and  6,  when  ft,  w,  and  t  are  given.     Since 


84  THEORETICAL   ASTRONOMY. 

cos  b  is  always  positive,  it  follows  that  I  —  &  and  u  must  lie  in  the 
same  quadrant  when  i  is  less  than  90° ;  but  if  i  is  greater  than  90°, 
or  the  motion  is  retrograde,  I  —  ft  and  360°  —  u  will  belong  to  the 
same  quadrant.  Hence  the  ambiguity  which  the  determination  of 
/  —  ft  by  means  of  its  tangent  involves,  is  wholly  avoided. 

If  we  use  the  distinction  of  retrograde  motion,  and  consider  i 
always  less  than  90°,  I —  ft  and  — u  will  lie  in  the  same  quadrant. 

32.  By  multiplying  the  first  of  the  equations  (81)  by  sin  u,  and 
the  second  by  cos  u,  and  combining  the  results,  considering  only  the 
upper  sign,  we  derive 

cos  b  sin  (u  —  I  -f-  ft  )  =  2  sin  u  cos  u  sin'  ^*, 
or 

cos  b  sin  (u  —  I  +  ft  )  =  sin  2u  sin8  \i. 

In  a  similar  manner,  we  find 

cos  b  cos  (u  —  l-\-  ft  )  =  cos'it  -J-  sin2w  cos  i, 
which  may  be  written 

cos  b  cos  (u  — 1-\-  ft  )  =  \  (1  -f-  cos  2-w)  +  J  (1  —  cos  2w)  cos  i, 
or 

cos  b  cos  (u  —  l-\-  ft  )  =  £(1  +  cost)  -j-  i  (1 —  cos  i)  cos  2u; 

and  hence 

cos  b  cos  (M  —  /  -f~  ft  )  =  cos7  £i  -f-  sin*  Ji  cos  2u. 

If  we  divide  this  equation  by  the  value  of  cos  b  sin  (u  —  I  +  ft ) 
already  found,  we  shall  have 

f        7  *    ^\          tan*J*sin2ti  ,QO, 

tan  (u  —  I  +  ft)  =  r—    L  2, . . JT-.  (83) 

1  -f-  tan2  £i  cos  2it 

The  angle  u  —  1+  ft  is  called  the  reduction  to  the  ecliptic;  and  the 
expression  for  it  may  be  arranged  in  a  series  which  converges  rapidly 
when  i  is  small,  as  in  the  case  of  the  planets.  In  order  to  effect  this 
development,  let  us  first  take  the  equation 

n  sin.  a; 

tan  v  =  j — : 

1  -f-  n  cos  x 

Differentiating  this,  regarding  y  and  n  as  variables,  and  reducing,  we 

find 

dy sin  a; 

~dn      1  -J-  2n  cos  x  -f-  ^ 


POSITION   IN   SPACE.  85 

which  gives,  by  division,  or  by  the  method  of  indeterminate  coefficients, 

-~  =  sin  x  —  n  sin  2x  +  n2  sin  3#  —  n3  sin  4#  -4-  &c. 
dn 

Integrating  this  expression,  we  get,  since  y  =  0  when  x  =  0, 

y  =  n  sin  x  —  \r?  sin  2#  -f-  |n3  sin  3#  —  \n^  sin  4#  -f-  ____  ,       (84) 

which  is  the  general  form  of  the  development  of  the  above  expression 
for  tan  y.  The  assumed  expression  for  tan  y  corresponds  exactly  with 
the  formula  for  the  reduction  to  the  ecliptic  by  making  n  =  tan2  Jt 
and  x  =  2u;  and  hence  we  obtain 

u  —  I  -j-  &  =  tan2  \i  sin  2u  —  ^  tan4  £i  sin  4^  -f~  i  tan6  Ji  sin  Qu 

—  &c.    '  (85) 


When  the  value  of  i  does  not  exceed  10°  or  12°,  the  first  two  terms 
of  this  development  will  be  sufficient.  To  express  u  —  I  -f-  &  in 
seconds  of  arc,  the  value  derived  from  the  second  member  of  this 
equation  must  be  multiplied  by  206264.81,  the  number  of  seconds 
corresponding  to  the  radius  of  a  circle. 

If  we  denote  by  jRe  the  reduction  to  the  ecliptic,  we  shall  have 


But  we  have  v  =  M  -f  the  equation  of  the  centre  ;  hence 

1  =  M  -f  TT  -f-  equation  of  the  centre  —  reduction  to  the  ecliptic, 
and,  putting  L  =  M  -f-  TT  =  mean  longitude,  we  get 

I  —  L  -f-  equation  of  centre  —  reduction  to  ecliptic.  (86) 

In  the  tables  of  the  motion  of  the  planets,  the  equation  of  the 
centre  (53)  is  given  in  a  table  with  M  as  the  argument  ;  and  the 
reduction  to  the  ecliptic  is  given  in  a  table  in  which  i  and  u  are  the 
arguments. 

33.  In  determining  the  place  of  a  heavenly  body  directly  from 
the  elements  of  its  orbit,  there  will  be  no  necessity  for  computing  the 
reduction  to  the  ecliptic,  since  the  heliocentric  longitude  and  latitude 
may  be  readily  found  by  the  formulae  (82).  When  the  heliocentric 
place  has  been  found,  we  can  easily  deduce  the  corresponding  geo- 
centric place. 

Let  x,  y,  z  be  the  rectangular  co-ordinates  of  the  planet  or  comet 
referred  to  the  centre  of  the  sun,  the  plane  of  xy  being  in  the  ecliptic, 


86  THEORETICAL   ASTRONOMY. 

the  positive  axis  of  x  being  directed  to  the  vernal  equinox,  and  the 
positive  axis  of  z  to  the  north  pole  of  the  ecliptic.  Then  we  shall 

have 

x  =  r  cos  b  cos  I, 
y  =  r  cos  b  sin  I, 
2  =r  sin  b. 

Again,  let  Xy  F,  Z  be  the  co-ordinates  of  the  centre  of  the  sun  re- 
ferred to  the  centre  of  the  earth,  the  plane  of  XY  being  in  the  eclip- 
tic, and  the  axis  of  X  being  directed  to  the  vernal  equinox  ;  and  let 
0  denote  the  geocentric  longitude  of  the  sun,  R  its  distance  from 
the  earth,  and  I  its  latitude.  Then  we  shall  have 

X=R  cos  S  cos  O, 
Y=  R  cos  2  sin  O, 
Z  =  R  sin  2. 

Let  x'j  yf,  z'  be  the  co-ordinates  of  the  body  referred  to  the  centre  of 
the  earth  ;  and  let  X  and  ft  denote,  respectively,  the  geocentric  longi- 
tude and  latitude,  and  J,  the  distance  of  the  planet  or  comet  from  the 
earth.  Then  we  obtain 

xf  =  A  cos  ft  cos  A, 

yf=Jco8^Bin^  (87) 

z'  =  A  sin  p. 

But,  evidently,  we  also  have 


and,  consequently, 

A  cos  p  cos  A  =  r  cos  b  cos  l-{-  R  cos  2  cos  O  > 

A  cos  p  sin  A  =  r  cos  6  sin  I  -J-  .R  cos  2"  sin  O  >  (88) 

J  sin  /?          =  r  sin  b          -\-  R  sin  2". 

If  we  multiply  the  first  of  these  equations  by  cos  Q,  and  the  second 
by  sin  Q,  and  add  the  products;  then  multiply  the  first  by  sin  Q, 
and  the  second  by  cos  O  ,  and  subtract  the  first  product  from  the 
second,  we  get 

A  cos  p  cos  (A  —  Q)  =  r  cos  b  cos  (I  —  Q)  -J-  R  cos  I, 

A  cos/3sm(A  —  0)  =  rcos6sin(J  —  O),  (89) 

A  sin  p  =  r  sin  b  -j-  R  sin  2". 

It  will  be  observed  that  this  transformation  is  equivalent  to  the  sup- 
position that  the  axis  of  x,  in  each  of  the  co-ordinate  systems,  is 


POSITION   IN   SPACE.  87 

directed  to  a  point  whose  longitude  is  ©>  or  that  the  system  has  been 
revolved  about  the  axis  of  z  to  a  new  position  for  which  the  axis  of 
abscissas  makes  the  angle  Q  with  that  of  the  primitive  system.  We 
may,  therefore,  in  general,  in  order  to  effect  such  a  transformation  in 
systems  of  equations  thus  derived,  simply  diminish  the  longitudes  by 
the  given  angle. 

The  equations  (89)  will  determine  A,  /9,  and  A  when  r.  6.  and  I  have 
been  derived  from  the  elements  of  the  orbit,  the  quantities  J?,  ©,  and 
Z  being  furnished  by  the  solar  tables;  or,  when  J,  /9,  and  A  are  given, 
these  equations  determine  /,  6,  and  r.  The  latitude  21  of  the  sun 
never  exceeds  ±  0".9,  and,  therefore,  it  may  in  most  cases  be  neg- 
lected, so  that  cos  £  —  1  and  sin  £  =  0,  and  the  last  equations  become 

A  cos  ft  cos  0*  —  Q  )  =  r  cos  b  cos  (I  —  0  )  -j-  R, 

A  cos/5  sin  (A  —  0)  =r  cosb  sin  (I—  ©),  (90) 

A  sin  /?  =r  sin  b. 

If  we  suppose  the  axis  of  x  to  be  directed  to  a  point  whose  longi- 
tude is  &,  or  to  the  ascending  node  of  the  planet  or  comet,  the  equa- 
tions (88)  become 

A  cos/?  cos  (A  —  &)  =r  cos  it  -f-  ^  cos -T  cos  (O  —  &), 

A  cos/?  sin  (A  —  Q)  =  ±r  smu  cosi  +  R  cos  2  sin  (0  —  &),  (91) 

A  sin  /?  =       r  sin  u  sin  i  -f-  R  sin  £, 

by  means  of  which  /9  and  ^  may  be  found  directly  from  & ,  i,  r,  and  u. 
If  it  be  required  to  determine  the  geocentric  right  ascension  and 
declination,  denoted  respectively  by  a  and  d,  we  may  convert  the 
values  of  f)  and  A  into  those  of  a  and  8.  To  effect  this  transforma- 
tion, denoting  by  e  the  obliquity  of  the  ecliptic,  we  have 

cos  8  cos  a  =  cos  /?  cos  A, 

cos  5  sin  a  =  cos  /?  sin  A  cos  e  —  sin  0  sin  e, 

sin  d  =  cos  /5  sin  A  sin  e  -f-  sin  /5  cos  e. 

Let  us  now  take 

n  sin  JV=  sin/?, 

w  cos  JV=  cos  /?  sin  A, 

and  we  shall  have 

COS  d  COS  tt  =  COS  /?  COS  A, 

cos  5  sin  a  =  n  cos  (JV+  0> 
sin  d  =n  sin  ( JV  -j-  €)» 


88  THEORETICAL,   ASTRONOMY. 

Therefore,  we  obtain 

(92) 


cos  N 
tan  5  =  tan  (N  +  e)  sin  a. 

We  also  have 

cos  (N  -f-  e)  _  cos  <5  sin  a 
cos  N  cos  /?  sin  A  ' 

which  will  serve  to  check  the  calculation  of  a  and  d.  Since  ccs  d  ana 
cos  /?  are  always  positive,  cos  a  and  cos  A  must  have  the  same  sign, 
and  thus  the  quadrant  in  which  oc  is  to  be  taken,  is  determined. 

For  the  solution  of  the  inverse  problem,  in  which  cc  and  d  are 
given  and  the  values  of  X  and  /?  are  required,  it  is  only  necessary  to 
interchange,  in  these  equations,  a  and  ^,  3  and  /?,  and  to  write  —  s  in 
place  of  e. 

34.  Instead  of  pursuing  the  tedious  process,  when  several  places 
are  required,  of  computing  first  the  heliocentric  place,  then  the  geo- 
centric place  referred  to  the  ecliptic,  and,  finally,  the  geocentric  right 
ascension  and  declination,  we  may  derive  formulae  which,  when  cer- 
tain constant  auxiliaries  have  once  been  computed,  enable  us  to  derive 
the  geocentric  place  directly,  referred  either  to  the  ecliptic  or  to  the 
equator. 

We  will  first  consider  the  case  in  which  the  ecliptic  is  taken  as  the 
fundamental  plane.  Let  us,  therefore,  resume  the  equations 

x'  =  r  cos  u, 

y'  =  d=  r  sin  u  cos  i, 

z'  =  r  sin  u  sin  i, 

in  which  the  axis  of  x  is  supposed  to  be  directed  to  the  ascending  node 
of  the  orbit  of  the  body.  If  we  now  pass  to  a  new  system  x,  y,  z,  — 
the  origin  and  the  axis  of  z  remaining  the  same,  —  in  which  the  axis 
of  x  is  directed  to  the  vernal  equinox,  we  shall  move  it  back,  in  a 
negative  direction,  equal  to  the  angle  &,  and,  consequently, 


x  =  xf  cos  R>  —  if  sin 
y  =  x'  sin  Q  -}-  y'  cos 


Therefore,  we  obtain 


fc  =  r  (cos  u  cos  &  =P  sin'it  cos  i  sin  &  ), 

y  =  r  (±  sin  u  cos  i  cos  &  -}-  cos  u  sin  &  ),  (93) 

z  =  r  sin  u  sin  i, 


POSITION   IN  SPACE.  89 

which  are  the  expressions  for  the  heliocentric  co-ordinates  of  a  planet 
or  comet  referred  to  the  ecliptic,  the  positive  axis  of  x  being  directed 
to  the  vernal   equinox.     The   upper  sign  is  to  be  used  when   the 
motion  is  direct,  and  the  lower  sign  when  it  is  retrograde. 
Let  us  now  put 

cos  Q  =  sin  a  sin  A, 

^F  cos  i  sin  &  =  sin  a  cos  A, 

sin  a  =  smbsmB, 

±  cos  i  cos  &  =  sin  b  cos  B, 

in  which  sin  a  and  sin  6  are  positive,  and  the  expressions  for  the  co- 
ordinates become 

x  =  r  sin  a  sin  (J.  -f-  u)t 

y  =  rsmb  sin  (B  -f  w),  (95) 

z  =r  sin  i  sin  it. 

The  auxiliary  quantities  a,  6,  J.,  and  .B,  it  will  be  observed,  are 
functions  of  &  and  i9  and,  in  computing  an  ephemeris,  are  constant 
so  long  as  these  elements  are  regarded  as  constant.  They  are  called 
the  constants  for  the  ecliptic. 

To  determine  them,  we  have,  from  equations  (94), 

cot  A  =  q=  tan  &  cos  i,  cot  B  —  ±  cot  Q>  cos  i, 

cos  &  .    ,       sin  & 

sma=  — — -p  sm0  =  — — ^-; 

sinJ.  sin  li 

the  upper  sign  being  used  when  the  motion  is  direct,  and  the  lower 
sign  when  it  is  retrograde. 

The  auxiliaries  sin  a  and  sin  b  are  always  positive,  and,  therefore, 
sin  A  and  cos  & ,  sin  B  and  sin  & ,  respectively,  must  have  the  same 
signs.  The  quadrants  in  which  A  and  B  are  situated,  are  thus  deter- 
mine^. 

From  the  equations  (94)  we  easily  find 

cos  a  =  sin  i  sin  &  > 

cos  b  =  —  sin  t  cos  & .  (96) 

If  we  add  to  the  heliocentric  co-ordinates  of  the  body  the  co-ordi- 
nates of  the  sun  referred  to  the  earth,  for  which  the  equations  have 
already  been  given,  we  shall  have 

x  -j-  X=  A  cos  ft  cos  A, 

y+  Y=  J  cos /9  sin  A,  (97) 

2  +  Z  =  A  sin  /?, 


90  THEORETICAL   ASTRONOMY. 

which  suffice  to  determine  A,  /?,  and  J.  The  values  of  a  and  8  may 
be  derived  from  these  by  means  of  the  equations  (92). 

35.  "We  shall  now  derive  the  formulae  for  determining  a  and  d 
directly.  For  this  purpose,  let  #,  y,  z  be  the  heliocentric  co-ordinates 
of  the  body  referred  to  the  equator,  the  positive  axis  of  x  being 
directed  to  the  vernal  equinox.  To  pass  from  the  system  of  co- 
ordinates referred  to  the  ecliptic  to  those  referred  to  the  equator  as 
the  fundamental  plane,  we  must  revolve  the  system  negatively  around 
the  axis  of  x,  so  that  the  axes  of  z  and  y  in  the  new  system  make 
the  angle  e  with  those  of  the  primitive  system,  e  being  the  obliquity 
of  the  ecliptic.  In  this  case,  we  have 

~" ~ 

X     X, 

2/"  =  y  cos  s  —  z  sin  e, 
z"  =  y  sin  e  -j-  z  cos  e. 

Substituting  for  x,  y,  and  z  their  values  from  equations  (93),  and 
omitting  the  accents,  we  get 

x  =  r  cos  u  cos  &  qc  r  sin  u  cos  i  sin  & , 

y  —  r  cos  u  sin  &  cos  s-\-r  sin  u  (±  cos  i  cos  &  cos  e — sin  i  sin  e),     (98) 

z  =  r  cos  u  sin  &  sin  e-\-r  sin  u  (db  cos  i  cos  &  sin  e  -j-  sin  i  cos  e). 

These  are  the  expressions  for  the  heliocentric  co-ordinates  of  the 
planet  or  comet  referred  to  the  equator.  To  reduce  them  to  a  con- 
venient form  for  numerical  calculation,  let  us  put 

cos  &  =  sin  a  sin  A, 

qp  cos  i  sin  &  =  sin  a  cos  A, 

sin  &  cos  e  =  sin  b  sin  i?, 

±  cos  i  cos  &  cos  e  —  sin  i  sin  e  =  sin  b  cos  -B, 

sin  £2  sin  e  =  sin  c  sin  0, 

±  cos  i  cos  &  sin  e  -f  sin  i  cos  e  =  sin  c  cos  (7; 

and  the  expressions  for  the  co-ordinates  reduce  to 

x  =  r  sin  a  sin  (A  -f-  u), 

y  =  rsmb  sin  (.£  -f  u),  (100) 

2  =  r  sin  c  sin  ( (7  -f-  u). 

The  auxiliary  quantities,  a,  6,  c,  JL,  jB,  and  C,  are  constant  so  long 
as  &  and  i  remain  unchanged,  and  are  called  constants  for  th(  equator. 

It  will  be  observed  that  the  equations  involving  a  and  A,  regard- 
ing the  motion  as  direct,  correspond  to  the  relations  between  the 
parts  of  a  quadrantal  triangle  of  which  the  sides  are  i  and  a,  the 


POSITION   IN   SPACE.  91 

angle  included  between  these  sides  being  that  which  we  designate  by 
A,  and  the  angle  opposite  the  side  a  being  90°  —  Q> .  In  the  case 
of  b  and  jB,  the  relations  are  those  of  the  parts  of  a  spherical  triangle 
of  which  the  sides  are  6,  i,  and  90°  -f-  e,  B  being  the  angle  included 
by  i  and  6,  and  180°  —  &  the  angle  opposite  the  side  b.  Further, 
in  the  case  of  c  and  (7,  the  relations  are  those  of  the  parts  of  a 
spherical  triangle  of  which  the  sides  are  c,  i,  and  e,  the  angle  C  being 
that  included  by  the  sides  i  and  c,  and  180°  —  &  that  included  by 
the  sides  i  and  e.  We  have,  therefore,  the  following  additional 
equations : 

cos  a  =  sin  i  sin  & , 

cos  b  =  —  cos  &  sin  i  cos  s  —  cos  i  sin  e,  (101) 

cos  e  =  —  cos  Q  sin  i  sin  s  -f-  cos  i  cos  e. 

In  the  case  of  retrograde  motion,  we  must  substitute  in  these 
180°  —  i  in  place  of  i. 

The  geometrical  signification  of  the  auxiliary  constants  for  the 
equator  is  thus  made  apparent.  The  angles  a,  b,  and  c  are  those 
which  a  line  drawn  from  the  origin  of  co-ordinates  perpendicular  to 
the  plane  of  the  orbit  on  the  north  side,  makes  with  the  positive  co- 
ordinate axes,  respectively ;  and  A,  B,  and  C  are  the  angles  which 
the  three  planes,  passing  through  this  line  and  the  co-ordinate  axes, 
make  with  a  plane  passing  through  this  line  and  perpendicular  to  the 
line  of  nodes. 

In  order  to  facilitate  the  computation  of  the  constants  for  the 
equator,  let  us  introduce  another  auxiliary  quantity  E0,  such  that 

sin  i  =  eQ  sin  EQ, 
±  cos  i  cos  Q  =  eQ  cos  Ew 

e(}  being  always  positive.     We  shall,  therefore,  have 

tani 
~  cos  Q,' 

Since  both  eQ  and  sini  are  positive,  the  angle  E0  cannot  exceed  180°; 
and  the  algebraic  sign  of  tan  EQ  will  show  whether  this  angle  is  to 
be  taken  in  the  first  or  second  quadrant. 
The  first  two  of  equations  (99)  give 

cot  A  =  HP  tan  &  cosi; 
and  the  first  gives 

cos  & 

sm  a  =  — — T. 
sin  A 


92  THEORETICAL   ASTRONOMY. 

From  the  fourth  of  equations  (99),  introducing  eQ  and  Eti,  we  get 

sin  b  cos  B  =  e0  cos  JE0  cos  e  —  e0  sin  EQ  sin  e  =  e0  cos  (J^  -{-  e)» 
But 

sin  6  sin  B  =  sin  &  cos  e ; 
therefore 


sin  &          cos  e  tan  &  cos  EQ  '        cos  e 

We  have,  also, 

.    ,       sin  0  cos  £ 

sin  b  = .    p     . 

sm.B 

In  a  similar  manner,  we  find 

cot  C=  ±        C°si  • sin  (^Q  +  e) 

~  tan  &  cos  EQ  '        sin  e 

and 

sin  O  sin  e 


sine  — 


sin  0 


The  auxiliaries  sin  a,  sin  6,  and  sin  c  are  always  positive,  and,  there- 
fore, sin  A  and  cos  &,  sini?  and  sin  &,  and  also  sin  Q  and  sin  &, 
must  have  the  same  signs,  which  will  determine  the  quadrant  in 
which  each  of  the  angles  A,  B,  and  C  is  situated. 

If  we  multiply  the  last  of  equations  (99)  by  the  third,  and  the 
fifth  of  these  equations  by  the  fourth,  and  subtract  the  first  product 
from  the  last,  we  get,  by  reduction, 

sin  b  sin  c  sin  (  C —  B)  =  —  sin  i  sin  &. 
But 

sin  a  cos  A  =  =F  cos  i  sin  &  J 


and  hence  we  derive 

sin  b  sin  c  sin  (  C —  B} 
sin  a  cos  A 


=  ±  tan  i, 


which  serves  to  check  the  accuracy  of  the  numerical  computation  of 
the  constants,  since  the  value  of  tan  i  obtained  from  this  formula 
must  agree  exactly  with  that  used  in  the  calculation  of  the  values  of 
these  constants. 

If  we  put  A'  =  A  ±  n  T  &,  B'  =  B  ±  n  q=  a,  and  C'  =  C±  n 
^F  & ,  the  upper  or  lower  sign  being  used  according  as  the  motion  is 
direct  or  retrograde,  we  shall  have 


POSITION  IN  SPACE.  93 

«  =  r  sin  a  sin  (A'  -f-  v), 

y  =  r  sin  5  sin  (£'  -f  *),  (102) 

z  =  r  sin  c  sin  ( C'  +  v), 

a  transformation  which  is  perhaps  unnecessary,  but  which  is  con- 
venient when  a  series  of  places  is  to  be  computed. 

It  will  be  observed  that  the  formula  for  computing  the  constants 
a,  6,  c,  A,  By  and  (7,  in  the  case  of  direct  motion,  are  converted  into 
those  for  the  case  in  which  the  distinction  of  retrograde  motion  id 
adopted,  by  simply  using  180°  — i  instead  of  i. 

36.  When  the  heliocentric  co-ordinates  of  the  body  have  been 
found,  referred  to  the  equator  as  the  fundamental  plane,  if  we  add  to 
these  the  geocentric  co-ordinates  of  the  sun  referred  to  the  same 
fundamental  plane,  the  sum  will  be  the  geocentric  co-ordinates  of 
the  body  refeired  also  to  the  equator. 

For  the  co  ordinates  of  the  sun  referred  to  the  centre  of  the  earth, 
we  have,  neglecting  the  latitude  of  the  sun, 

X=  RcosQ, 

Y=- .Rsin  O  cose, 

Z  =  R  sin  O  sin  e  =  Ftan  e, 

in  which  R  represents  the  radius-vector  of  the  earth,  O  the  sun's 
longitude,  and  e  the  obliquity  of  the  ecliptic. 
We  shall,  therefore,  have 

x  -}-  JT  =  A  cos  £  cos  a, 

y  -f  Y=  A  cos  8  sin  a,  (103) 

z -f  Z=  Jsind, 

which  suffice  to  determine  a,  d,  and  J. 

If  we  have  regard  to  the  latitude  of  the  sun  in  computing  its  geo- 
centric co-ordinates,  the  formulae  will  evidently  become 

JT—  Ii  cos  O  cos  S, 

Y=  R  sin  Q  cos  S cos  e  —  R  sin  2 sine,  (104) 

Z  =  H  sin  O  cos  £  sin  e  -\-  JK  sin  S  cos  e, 

in  which,  since  S  can  never  exceed  it  0".9,  cos  2  is  very  nearly 
equal  to  1,  and  sin  I  =  I. 

The  longitudes  and  latitudes  of  the  sun  may  be  derived  from  a 
solar  ephemeris,  or  from  the  solar  tables.  The  principal  astronomical 
ephemerides,  such  as  the  Berliner  Astronomisches  Jahrbuch,  the 
Nautical  Almanac,  and  the  American  Ephemeris  and  Nautical  Al- 


94  THEOKETICAL  ASTRONOMY. 

manac,  contaiD,  for  each  year  for  which  they  are  published,  the 
equatorial  co-ordinates  of  the  sun,  referred  both  to  the  mean  equinox 
and  equator  of  the  beginning  of  the  year,  and  to  the  apparent  equinox 
of  the  date,  taking  into  account  the  latitude  of  the  sun. 

37.  In  the  case  of  an  elliptic  orbit,  we  may  determine  the  co- 
ordinates directly  from  the  eccentric  anomaly  in  the  following 
manner  :  — 

The  equations  (102)  give,  accenting  the  letters  a,  6,  and  c, 

x  =  r  cos  v  sin  a'  sin  A'  -j-  r  sin  v  sin  a'  cos  A', 
y  =r  cos  v  sin  br  sin  Br  -f-  r  sin  v  sin  b'  cos  B', 
z  =  r  cos  v  sin  cr  sin  C'  -[-r  sin  v  sin  c'  cos  C'. 

Now,  since  r  cos  v  =  a  cos  E  —  ae,  and  r  sin  v  =  a  cos  <p  sin  E,  we  shall 
have 

x  —  a  sin  a'  sin  J/  cos  E  —  ae  sin  a'  sin  Af  -}-  a  cos  ^  sin  a'  cos  J'  sin  .7?, 
y  =  a  sin  5'  sin  B'  cos  J?  —  ae  sin  6'  sin  B'  -\-  a  cos  p  sin  6'  cos  B'  sin  jE7, 
2  =  a  sin  c'  sin  C"  cos  E  —  ae  sin  c'  sin  C"  -f  a  cos  ^  sin  c'  cos  C^  sin  E. 

Let  us  now  put 

a  cos  ^  sin  a'  cos  A'  =  Ax  cos  Lv 
a  sin  a'  sin  A'  =  Ax  sin  Zrx, 

—  ae  sin  a'  sin  A'  =  —  eAx  sin  jkx  =  vx  ; 
a  cos  <p  sin  5'  cos  1?'  =  ^  cos  _Ly, 

a  sin  V  sin  5'  =  ^y  sin  JrT, 

—  ae  sin  bf  sin  j8'  =  —  eAy  sin  L7  =  v7; 
a  cos  p  sin  c'  cos  C'  =  lz  cos  jLz, 

a  sin  c'  sin  C"  =  ^,  sin  Z/,, 

—  ae  sin  c'  sin  0'  =  —  e*t  sin  JD,  =  v,  ; 

in  which  sin  a',  sin  6',  and  sin  c'  have  the  same  values  as  in  equations 
(102),  the  accents  being  added  simply  to  mark  the  necessary  dis- 
tinction in  the  notation  employed  in  these  formulae.  "We  shall, 

therefore,  have 

z=:/lxsm(£x+.E)-i-vx, 

y=aysin(J-hJB)  +  vy,  (105) 


By  means  of  these  formula?,  the  co-ordinates  are  found  directly 
from  the  eccentric  anomaly,  when  the  constants  ^x,  ^,,  ^z,  Xx,  Ly,  Z/,, 
vx,  vy,  and  vz,  have  been  computed  from  those  already  found,  or  from 
a,  b}  c,  Ay  jB,  and  C.  This  method  is  very  convenient  when  a  great 


POSITION   IN   SPACE.  95 

number  of  geocentric  places  are  to  be  computed ;  but,  when  only  a 
few  places  are  required,  the  additional  labor  of  computing  so  many 
auxiliary  quantities  will  not  be  compensated  by  the  facility  afforded 
in  the  numerical  calculation,  when  these  constants  have  been  deter- 
mined. Further,  when  the  ephemeris  is  intended  for  the  comparison 
of  a  series  of  observations  in  order  to  determine  the  corrections  to  be 
applied  to  the  elements  by  means  of  the  differential  formulae  which 
we  shall  investigate  in  the  following  chapter,  it  will  always  be  ad- 
visable to  compute  the  co-ordinates  by  means  of  the  radius-vector 
and  true  anomaly,  since  both  of  these  quantities  will  be  required  in 
finding  the  differential  coefficients. 

38.  In  the  case  of  a  hyperbolic  orbit,  the  co-ordinates  may  be  com- 
puted directly  from  F,  since  we  have 

r  cos  v  =  a  (e  —  sec  jP), 
r  sin  v  =  a  tan  4  tan  F ; 
and,  consequently, 

x  =  ae  sin  a'  sin  A'  —  a  sec  F  sin  a!  sin  A  -\- a  tan  4  tan  F  sin  a'  cos  A', 
y  =  ae  sin  bf  sin  B'  —  a  sec  F  sin  b'  sin  B'  -f  a  tan  4-  tan  F  sin  V  cos  Bft 
z  =  ae  sin  c'  sin  C'  —  a  sec  F  sin  c'  sin  C"  -{-  a  tan  4*  tan  F  sin  c'  cos  C". 

Let  us  now  put 

ae  sin.  a'  sin  A  =  A*, 

—  a  sin  a'  sin  A'  =  ^ 
a  tan  4  sin  a!  cos  A'  =  vx ; 

ae  sin  £>'  sin  B'  =  Ay, 

—  a  sin  bf  sin  _B'  =  ^ty, 
a  tan  4>  sin  b'  cos  Bf  =  vy; 

ae  sin  c'  sin  C"  =  >*„ 

—  a  sin  c'  sin  Gr  =  /*„ 
a  tan  4  sin  c'  cos  C"  =  vz. 

Then  we  shall  have 

x  =  Ax  -f  fj.K  sec  .F  +  vx  tan  jP, 

v  =  Ay  4-  ^  sec  jF  +  vy  tan  F,  (106; 

2  =  As  -j-  /7.z  sec  F  -f-  vz  tan  jP. 

In  a  similar  manner  we  may  derive  expressions  for  the  co-ordinates, 
in  the  case  of  a  hyperbolic  orbit,  when  the  auxiliary  quantity  a  is 
used  instead  of  F. 

39.  If  we  denote  by  ;:',  £',  and  V  the  elements  which  determine 
the  position  of  the  orbit  in  space  when  referred  to  the  equator  as  the 


96  THEORETICAL   ASTRONOMY. 

fundamental  plane,  and  by  o)0  the  angular  distance  between  the 
ascending  node  of  the  orbit  on  the  ecliptic  and  its  ascending  node  on 
the  equator,  being  measured  positively  from  the  equator  in  the 
direction  of  the  motion,  we  shall  have 


To  find  Q,'  and  i',  we  have,  from  the  spherical  triangle  formed  by 
the  intersection  of  the  planes  of  the  orbit,  ecliptic,  and  equator  with 
the  celestial  vault, 

cos  if  =  cos  i  cos  e  —  sin  i  sin  e  cos  & , 
sin  ir  sin  &'  =  sin  i  sin  ^ , 
sin  i'  cos  &'  =  cos  t  sin  e  -f-  sin  i  cos  e  cos  SI  • 

Let  us  now  put 

n  sin  JV  =  cos  it 

n  cos  JV=  sin  i  cos  &, 

and  these  equations  reduce  to 

cos  i'  =  n  sin  ( JV  —  e), 
sin  i'  sin  &'  =  sin  i  sin  & , 
sin  i'  cos  &'  =  n  cos  (JV —  e)  ; 
from  which  we  find 

HT        cot  i  cos  JV 

tan  JV= — ,  tan  £'  = ^ ^  tan  &, 

cos  £1  cos  (JV  —  e) 

cot  i'  =  tan  (JV—  e)  cos  &'.  (107) 

Since  sin  i  is  always  positive,  cos  JV  and  cos  &  must  have  the  same 
signs.  To  prove  the  numerical  calculation,  we  have 

sin  i  cos  Q  _          cos  JV 
sin  i'  cos  ££'       cos  (JV —  e)' 


the  value  of  the  second  member  of  which  must  agree  with  that  used 
in  computing  & '. 

In  order  to  find  CDO,  we  have,  from  the  same  triangle, 

sin  CMO  sin  i'  =  sin  &  sin  e, 

cos  a>n  sin  i'  =  cos  e  sin  i  -{-  sin  e  cos  i  cos  & . 

Liet  us  now  take 

m  sin  Jlf  =  cos  e, 

m  cos  Jlf  =  sin  e  cos  &  ; 
and  we  obtain 


POSITION   IN   SPACE.  97 

cot  M  =  tan  e  cos  &, 


and,  also,  to  check  the  calculation, 

sin  e  cos  &  cos  Jf 


sin  i'  cos  %       cos  (M —  i)' 

If  we  apply  Gauss's  analogies  to  the  same  spherical  triangle,  we 
get 

cos^'  cos^  (£'  +  «>0)  ==  cosi£  cosi  (*'  +  s\  MAQx 

sin  It  sin  I  (&'  —  «0)  —  sin  J&  sin  i  (t  —  e), 


The  quadrant  in  which  \  (&'  +  ^0)  or  K^  —  ^o)  ^s  situated,  must  be 
so  taken  that  sin  \if  and  cos  \V  shall  be  positive  ;  and  the  agreement 
of  the  values  of  the  latter  two  quantities,  computed  by  means  of  the 
value  of  \if  derived  from  tan  Ji',  will  serve  to  check  the  accuracy  of 
the  numerical  calculation. 

For  the  case  in  which  the  motion  is  regarded  as  retrograde,  we 
must  use  180°  —  i  instead  of  i  in  these  equations,  and  we  have,  also, 


We  may  thus  find  the  elements  TT',  &  ',  and  ify  in  reference  to  the 
equator,  from  the  elements  referred  to  the  ecliptic;  and  using  the 
elements  so  found  instead  of  it,  &,  and  i,  and  using  also  the  places 
of  the  sun  referred  to  the  equator,  we  may  derive  the  heliocentric 
and  geocentric  places  with  respect  to  the  equator  by  means  of  the 
formulae  already  given  for  the  ecliptic  as  the  fundamental  plane. 

If  the  position  of  the  orbit  with  respect  to  the  equator  is  given, 
and  its  position  in  reference  to  the  ecliptic  is  required,  it  is  only 
necessary  to  interchange  &  and  &',  as  well  as  i  and  180°  —  i',  e 
remaining  unchanged,  in  these  equations.  These  formulae  may 
also  be  used  to  determine  the  position  of  the  orbit  in  reference  to 
any  plane  in  space;  but  the  longitude  &  must  then  be  measured 
from  the  place  of  the  descending  node  of  this  plane  on  the  ecliptic. 
The  value  of  &,  therefore,  which  must  be  used  in  the  solution  of  the 
equations  is,  in  this  case,  equal  to  the  longitude  of  the  ascending 
node  of  the  orbit  on  the  ecliptic  diminished  by  the  longitude  of  the 
descending  node  of  the  new  plane  of  reference  on  the  ecliptic.  The 
quantities  &  ',  i',  and  w0  will  have  the  same  signification  in  reference 

7 


98  THEORETICAL   ASTRONOMY. 

to  this  plane  that  they  have  in  reference  to  the  equator,  with  this  dis- 
tinction, however,  that  & '  is  measured  from  the  descending  node  of 
this  new  plane  of  reference  on  the  ecliptic ;  and  e  will  in  this  case 
denote  the  inclination  of  the  ecliptic  to  this  plane. 

40.  We  have  now  derived  all  the  formulae  which  can  be  required 
in  the  case  of  undisturbed  motion,  for  the  computation  of  the  helio- 
centric or  geocentric  place  of  a  heavenly  body,  referred  either  to  the 
ecliptic  or  equator,  or  to  any  other  known  plane,  when  the  elements 
of  its  orbit  are  known ;  and  the  formulae  which  have  been  derived 
are  applicable  to  every  variety  of  conic  section,  thus  including  all 
possible  forms  of  undisturbed  orbits  consistent  with  the  law  of  uni- 
versal gravitation.  The  circle  is  an  ellipse  of  which  the  eccentricity 
is  zero,  and,  consequently,  M=v  =  u,  and  r  =  a,  for  every  point  of 
the  orbit.  There  is  no  instance  of  a  circular  orbit  yet  known ;  but 
in  the  case  of  the  discovery  of  the  asteroid  planets  between  Mars 
and  Jupiter  it  is  sometimes  thought  advisable,  in  order  to  facilitate 
the  identification  of  comparison  stars  for  a  few  days  succeeding  the 
discovery,  to  compute  circular  elements,  and  from  these  an  ephemeris. 

The  elements  which  determine  the  form  of  the  orbit  remain  con- 
stant so  long  as  the  system  of  elements  is  regarded  as  unchanged ; 
but  those  which  determine  the  position  of  the  orbit  in  space,  TT,  & , 
and  i,  vary  from  one  epoch  to  another  on  account  of  the  change  of 
the  relative  position  of  the  planes  to  which  they  are  referred.  Thus 
the  inclination  of  the  orbit  will  vary  slowly,  on  account  of  the  change 
of  the  position  of  the  ecliptic  in  space,  arising  from  the  perturbations 
of  the  earth  by  the  other  planets ;  while  the  longitude  of  the  peri- 
helion and  the  longitude  of  the  ascending  node  will  vary,  both  on 
account  of  this  change  of  the  position  of  the  plane  of  the  ecliptic, 
and  also  on  account  of  precession  and  nutation.  If  TT,  &,  and  i  are 
referred  to  the  true  equinox  and  ecliptic  of  any  date,  the  resulting 
heliocentric  places  will  be  referred  to  the  same  equinox  and  ecliptic ; 
and,  further,  in  the  computation  of  the  geocentric  places,  the  longi- 
tudes of  the  sun  must  be  referred  to  the  same  equinox,  so  that  the 
resulting  geocentric  longitudes  or  right  ascensions  will  also  be  re- 
ferred to  that  equinox.  It  will  appear,  therefore,  that,  on  account 
of  these  changes  in  the  values  of  n,  &,  and  i,  the  auxiliaries  sin  a, 
sin  6,  sin  c,  A,  J5,  and  C,  introduced  into  the  formulae  for  the  co- 
ordinates, will  not  be  constants  in  the  computation  of  the  places  for 
a  series  of  dates,  unless  the  elements  are  referred  constantly,  in  the 
calculation,  to  a  fixed  equinox  and  ecliptic.  It  is  customary,  there- 


POSITION   IN   SPACE.  99 

fore,  to  reduce  the  elements  to  the  ecliptic  and  mean  equinox  of  the 
beginning  of  the  year  for  which  the  ephemeris  is  required,  and  then 
to  compute  the  places  of  the  planet  or  comet  referred  to  this  equinox, 
using,  in  the  case  of  the  right  ascension  and  declination,  the  mean 
obliquity  of  the  ecliptic  for  the  date  of  the  fixed  equinox  adopted,  in 
the  computation  of  the  auxiliary  constants  and  of  the  co-ordinatey 
of  the  sun.  The  places  thus  found  may  be  reduced  to  the  true 
equinox  of  the  date  by  the  well-known  formulae  for  precession  and 
nutation.  Thus,  for  the  reduction  of  the  right  ascension  and  declina- 
tion from  the  mean  equinox  and  equator  of  the  beginning  of  the 
year  to  the  apparent  or  true  equinox  and  equator  of  any  date,  usually 
the  date  to  which  the  co-ordinates  of  the  body  belong,  we  have 

A<*  =/+  g  sin  (G  +  a)  tan  d, 


for  which  the  quantities  /,  g,  and  G  are  derived  from  the  data  given 
either  in  the  solar  and  lunar  tables,  or  in  astronomical  ephemerides, 
such  as  have  already  been  mentioned. 

The  problem  of  reducing  the  elements  from  the  ecliptic  of  one 
date  t  to  that  of  another  date  tf  may  be  solved  by  means  of  equations 
(109),  making,  however,  the  necessary  distinction  in  regard  to  the 
point  from  which  Q>  and  &>'  are  measured.  Let  d  denote  the  longi- 
tude of  the  descending  node  of  the  ecliptic  of  t'  on  that  of  ty  and 
let  T}  denote  the  angle  which  the  planes  of  the  two  ecliptics  make 
with  each  other,  then,  in  the  equations  (109),  instead  of  &  we  must 
write  &  —  0,  and,  in  order  that  Q,  '  shall  be  measured  from  the 
vernal  equinox,  we  must  also  write  &  '  —  6  in  place  of  Q,  '  .  Finally, 
we  must  write  T]  instead  of  e,  and  A<w  for  o)w  which  is  the  variation 
in  the  value  of  a)  in  the  interval  tf  —  t  on  account  of  the  change  of 
the  position  of  the  ecliptic  ;  then  the  equations  become 


nj(£'  —  8+  AW)  =  sini(&  —  0}  cos  £  (i  —  ^)> 
os£  (£'—  0  +  Aw)  =  cosi  (&  —  0)  cos£  (i  +  17), 
sin  Jt*  sin  J  (£'  —  0  —  Aw)  =  sin  J  (&  —  0)  sin  J  (i  —  9), 
sin  $i!  cosi  (£'  —  0  —  AW)  =  Cos£  (&  —  0)  sin  J  (t  -f  r,). 


These  equations  enable  us  to  determine  accurately  the  values  of  &  ', 
i',  and  AW,  which  give  the  position  of  the  orbit  in  reference  to  the 
ecliptic  corresponding  to  the  time  tf,  when  d  and  y  are  known.  The 
longitudes,  however,  will  still  be  referred  to  the  same  mean  equinox 
as  before,  which  we  suppose  to  be  that  of  t;  and,  in  order  to  refer 


100  THEORETICAL   ASTRONOMY. 

them  to  the  mean  equinox  of  the  epoch  t',  the  amount  of  the  pre- 
cession in  longitude  during  the  interval  tf  —  t  must  also  be  applied. 

If  the  changes  in  the  values  of  the  elements  are  not  of  consider- 
able magnitude,  it  will  be  unnecessary  to  apply  these  rigorous  formulae, 
and  -we  may  derive  others  sufficiently  exact,  and  much  more  con- 
venient in  application.  Thus,  from  the  spherical  triangle  formed  by 
the  intersection  of  the  plane  of  the  orbit  and  of  the  planes  of  the 
two  ecliptics  with  the  celestial  vault,  we  get 

sin  7]  cos  (ft  —  0)  =  —  cos  i'  sin  i  -f-  sin  if  cos  i  cos  Aw, 
from  which  we  easily  derive 

sin  (ir  —  i)  =  sin  TJ  cos  (ft  —  0)  -f-  2  sin  if  cos  i  sin2  JAW.        (112) 
We  have,  further, 

sin  Aw  sin  i'  =  sin  f)  sin  (ft  —  0), 
or 

Shl(ft  -  0)  r+  +  n\ 

sin  AW  =  sm  -TJ  -  ^-3  —  -.  (113) 

sin* 

We  have,  also,  from  the  same  triangle, 

sin  AW  cos  i'  =  —  cos  (ft  —  0)  sin  (ft'  —  0) 

-f  sh  (ft  —  0)  cos  (ft'  —  0)  cos  17, 
which  givec 

sin  (ft'  —  ft)  =  —  sin  Aw  cos  if  —  2  sin  (ft  —  0)  cos  (ft'  —  0)  sin2  ^, 

or 

sin  (ft'  —  ft)  =  —  sin  >?  sin  (ft—  0)coti' 
—  2  sin  (ft  —  0)  cos  (ft'  —  0)  sin1  £7.  (114) 

Finally,  we  have 

TT'  —  TT=  ftf—  ft  +  Aw. 

Since  ^  is  very  small,  these  equations  give,  if  we  apply  also  the  pre- 
cession in  longitude  so  as  to  reduce  the  longitudes  to  the  mean  equinox 
of  the  date  *' 


smi 
if  =    i_-^cos(ft—  0)  - 


8 

2-0),      (115) 


POSITION   IN   SPACE.  101 

in  which  —  is  the  annual  precession  in  longitude,  and  in  which 

s  =  206264".  8.     In  most  cases,  the  last  terms  of  the  expressions  for 
if,  Q'y  and  TT',  being  of  the  second  order,  may  be  neglected. 

For  the  case  in  which  the  motion  is  regarded  as  retrograde,  we 
must  put  180°  —  i  and  180°  —  i',  instead  of  i  and  i',  respectively,  in 
the  equations  for  AO>,  if,  and  &  '  ;  and  for  TT',  in  this  case,  we  have 


which  gives 

*'  =  *  +  (*'  —  0-^7  —  7  sin  (&  —  0)  tan  £»'  —  ±^s 

Civ  S 

If  we  adopt  BesseFs  determination  of  the  luni-solar  precession  and 
of  the  variation  of  the  mean  obliquity  of  the  ecliptic,  we  have,  at  the 
time  1750  +  r, 

-^  =  50".21129  +  O."0002442966r, 
at 

^  =   0".48892  —  0/'000006143r, 
at 

and,  consequently, 

73  =  (0."48892  —  O."000006143r)  (f  —  f)  ; 

and  in  the  computation  of  the  values  of  these  quantities  we  must  put 
T  —  \(tr  4"  0  —  1750,  t  and  t1  being  expressed  in  years. 

The  longitude  of  the  descending  node  of  the  ecliptic  of  the  time  t 
on  the  ecliptic  of  1750.0  is  also  found  to  be 

351°  36'  10"  —  5".21  (t  —  1750), 

which  is  measured  from  the  mean  equinox  of  the  beginning  of  the  year 
1750. 

The  longitude  of  the  descending  node  of  the  ecliptic  of  tf  on  that 
of  tf  measured  from  the  same  mean  equinox,  is  equal  to  this  value 
diminished  by  the  angular  distance  between  the  descending  node  of 
the  ecliptic  of  t  on  that  of  1750  and  the  descending  node  of  the 
ecliptic  of  t'  on  that  of  t,  which  distance  is,  neglecting  terms  of  the 
second  order, 

5"31(f—  1750); 
and  the  result  is 

351°  36'  10"—  5".21  (t  —  1750)  —  5".21  (if—  1750), 
351°  36'  10"  -10".42(£  —  1750)—  5".21(/  —  *). 


102  THEORETICAL   ASTRONOMY. 

To  reduce  this  longitude  to  the  mean  equinox  at  the  time  t,  we  must 
add  the  general  precession  during  the  interval  t  —  1750,  or 

50".21  (t  — 1750), 
so  that  we  have,  finally, 

e  =  351°  36'  10"  +  39".79  (t  —  1750)  —  5".21  (if  —  <). 

When  the  elements  TT,  &,  and  i  have  been  thus  reduced  from  the 
ecliptic  and  mean  equinox  to  which  they  are  referred,  to  those  of  the 
date  for  which  the  heliocentric  or  geocentric  place  is  required,  they 
may  be  referred  to  the  apparent  equinox  of  the  date  by  applying  the 
nutation  in  longitude.  Then,  in  the  case  of  the  determination  of  the 
right  ascension  and  declination,  using  the  apparent  obliquity  of  the 
ecliptic  in  the  computation  of  the  co-ordinates,  we  directly  obtain  the 
place  of  the  body  referred  to  the  apparent  equinox.  But,  in  com- 
^uting  a  series  of  places,  the  changes  which  thus  take  place  in  the 
elements  themselves  from  date  to  date  induce  corresponding  changes 
in  the  auxiliary  quantities  a,  6,  c,  A,  B,  and  CJ  so  that  these  are  no 
longer  to  be  considered  as  constants,  but  as  continually  changing  their 
values  by  small  differences.  The  differential  formulae  for  the  com- 
putation of  these  changes,  which  are  easily  derived  from  the  equations 
(99),  will  be  given  in  the  next  chapter;  but  they  are  perhaps  unneces- 
sary, since  it  is  generally  most  convenient,  in  the  cases  which  occur,  to 
compute  the  auxiliaries  for  the  extreme  dates  for  which  the  ephemeris 
is  required,  and  to  interpolate  their  values  for  intermediate  dates. 

It  is  advisable,  however,  to  reduce  the  elements  to  the  ecliptic  and 
mean  equinox  of  the  beginning  of  the  year  for  which  the  ephemeris 
is  required,  and  using  the  mean  obliquity  of  the  ecliptic  for  that 
epoch,  in  the  computation  of  the  auxiliary  constants  for  the  equator, 
the  resulting  geocentric  right  ascensions  and  declinations  will  be 
referred  to  the  same  equinox,  and  they  may  then  be  reduced  to  the 
apparent  equinox  of  the  date  by  applying  the  corrections  for  preces- 
sion and  nutation. 

The  places  which  thus  result  are  free  from  parallax  and  aberration. 
In  comparing  observations  with  an  ephemeris,  the  correction  for  par- 
allax is  applied  directly  to  the  observed  apparent  places,  since  this 
correction  varies  for  different  places  on  the  earth's  surface.  The  cor- 
rection for  aberration  may  be  applied  in  two  different  modes.  We 
may  subtract  from  the  time  of  observation  the  time  in  which  the 
light  from  the  planet  or  comet  reaches  the  earth,  and  the  true  place 
for  this  reduced  time  is  identical  with  the  apparent  place  for  the  time 


NUMERICAL    EXAMPLES.  103 

of  observation ;  or,  in  case  we  know  the  daily  or  hourly  motion  of 
the  body  in  right  ascension  and  declination,  we  may  compute  the 
motion  during  the  interval  which  is  required  for  the  light  to  pass 
from  the  body  to  the  earth,  which,  being  applied  to  the  observed 
place,  gives  the  true  place  for  the  time  of  observation. 

We  may  also  include  the  aberration  directly  in  the  ephemeris  by 
using  the  time  t  —  497S.78^  in  computing  the  geocentric  places  for 
the  time  t,  or  by  subtracting  from  the  place  free  from  aberration,  com- 
puted for  the  time  t,  the  motion  in  a  and  d  during  the  interval 
497*.78  J,  in  which  expression  A  is  the  distance  of  the  body  from  tlu 
earth,  and  497.78  the  number  of  seconds  in  which  light  traverses  the 
mean  distance  of  the  earth  from  the  sun. 

It  is  customary,  however,  to  compute  the  ephemeris  free  from 
aberration  and  to  subtract  the  time  of  aberration,  497*.78  J,  from  the 
time  of  observation  when  comparing  observations  with  an  ephemeris, 
according  to  the  first  method  above  mentioned.  The  places  of  the 
sun  used  in  computing  its  co-ordinates  must  also  be  free  from  aberra- 
tion; and  if  the  longitudes  derived  from  the  solar  tables  include 
aberration,  the  proper  correction  must  be  applied,  in  order  to  obtain 
the  true  longitude  required. 

41.  EXAMPLES. — We  will  now  collect  together,  in  the  proper 
order  for  numerical  calculation,  some  of  the  principal  formulae  which 
have  been  derived,  and  illustrate  them  by  numerical  examples,  com- 
mencing with  the  case  of  an  elliptic  orbit.  Let  it  be  required  to  find 
the  geocentric  right  ascension  and  declination  of  the  planet  Eurynome 
©,  for  mean  midnight  at  Washington,  for  the  date  1865  February 
24,  the  elements  of  the  orbit  being  as  follows : — 

Epoch  =  1864  Jan.  1.0  Greenwich  mean  time. 
29'  40".21 

20  33  .09^    Ecliptic  and  Mean 
Equinox,  1864.0. 


log  a  =  0.3881319 
log  fJL  =  2.9678088 
P  =  928".55746 

When  a  series  of  places  is  to  be  computed,  the  first  thing  to  oe 
done  is  to  compute  the  auxiliary  constants  used  in  the  expressions  for 
the  co-ordinates,  and  although  but  a  single  pi  ice  is  required  in  the 
problem  proposed,  yet  we  will  proceed  in  this  manner,  in  order  to 


104  THEORETICAL   ASTRONOMY. 

exhibit  the  application  of  the  formulae.  Since  the  elements  it,  ££, 
and  i  are  referred  to  the  ecliptic  and  mean  equinox  of  1864.0,  we  will 
first  reduce  them  to  the  ecliptic  and  mean  equinox  of  1865.0.  For 
this  reduction  we  have  t  —  1864.0,  and  tf=  1865.0,  which  give 

rll 

5f  =  50".239,  9  =  352°  51'  41",  7  =  0".4882. 

at 

Substituting  these  values  in  the  equations  (115),  we  obtain 

if  —  i  =  At  =  —  0".40,  A&  =  +  53".61,  ATT  ^  +  50".23  ; 

and  hence  the  elements  which  determine  the  position  of  the  orbit  in 
reference  to  the  ecliptic  of  1865.0  are 

TT  =  44°  21'  23".32,  £  =  206°  43'  33".74,  i  =  4°  36'  50".ll. 

For  the  same  instant  we  derive,  from  the  American  Ephemeris  and 
Nautical  Almanac,  the  value  -of  the  mean  obliquity  of  the  ecliptic, 
which  is 

e  =  23°  27'  24".03. 

The  auxiliary  constants  for  the  equator  are  then  found  by  means  oi 
the  formulas 

cot  A  =  —  tan  &  cos  i,  tan  EQ  =  --  —  , 

COS  do 

cosi          c 


tan  &  cos  EQ          cos  e 
Coi0= 


tan  &  cos     Q          sin  e 

cos  Q>  .    ,        sin  &  cos  e  .  sin  &  sm  e 

sm  a  =  —  —  r,  sm  b  =  -  :  —  ^  —  ,  sm  c  —  -  :  —  ~  —  . 

sm  A  SIHJD  sm  C 

The  angle  E0  is  always  less  than  180°,  and  the  quadrant  in  which  it  is 
to  be  taken,  is  indicated  directly  by  the  algebraic  sign  of  tan  EQ.  The 
values  of  sin  a,  sin  6,  and  sin  c  are  always  positive,  and,  therefore,  the 
angles  A,  B,  and  C  must  be  so  taken,  with  respect  to  the  quadrant  in 
which  each  is  situated,  that  sin  A  and  cos  &,  sin  B  and  sin  &,  and  also 
sin  C  and  sin  &  ,  shall  have  the  same  signs.  From  these  we  derive 

A  =  296°  39'    5".07,  log  sin  a  =  9.9997156, 

B  =  205    55  27  .14,  log  sin  b  =  9.9748254, 

C  =  212    32  17  .74,  -  log  sin  c  =  9.5222192. 

Finally,  the  calculation  of  these  constants  is  proved  by  means  of  the 
formula 


NUMERICAL   EXAMPLES.  105 

.      sin  b  sin  c  sin  (  C —  -B) 

tan  i  = : — 

sin  a  cos  A 

whic;i  gives  log  tan  i  =  8.9068875,  agreeing  with  the  value  8.9068876 
derived  directly  from  i. 

Next,  to  find  r  and  u.  The  date  1865  February  24.5  mean  time 
at  Washington  reduced  to  the  meridian  of  Greenwich  by  applying 
the  difference  of  longitude,  5*  8m  1P.2,  becomes  1865  February 
24.714018  mean  time  at  Greenwich.  The  interval,  therefore,  from 
the  epoch  for  which  the  mean  anomaly  is  given  and  the  date  for 
which  the  geocentric  place  is  required,  is  420.714018  days;  and  mul- 
tiplying the  mean  daily  motion,  928". 55745,  by  this  number,  and 
adding  the  result  to  the  given  value  of  M,  we  get  the  mean  anomaly 
for  the  required  place,  or 

M  =  1°  29'  40".21  +  108°  30'  57".14  =  110°  0'  37".35. 
The  eccentric  anomaly  E  is  then  computed  by  means  of  the  equation 

M=E—  esmE, 

the  value  of  e  being  expressed  in  seconds  of  arc.  For  Eurynome  we 
have  log  sin  <p  =  log  e  =  9.2907754,  and  hence  the  value  of  e  ex- 
pressed in  seconds  is 

log  6  =  4.6052005. 

By  means  of  the  equation  (54)  we  derive  an  approximate  value  of  E, 

namely, 

4= 119°  49*  24", 

the  value  of  e2  expressed  in  seconds  being  log  e2=  3.895976;  and 
with  this  we  get 

MQ  =  EQ  —  e  sin  E^  =  110°  6'  50". 
Then  we  have 

M-JI.  372".7  .  , 

^^ItrjoiiE      'TOST- 

which  gives,  for  a  second  approximation  to  the  value  of  Et 

EQ  =  119°  43'  44".3. 
This  gives  M0=  110°  0'  36".98,  and  hence 


106  THEORETICAL   ASTRONOMY. 

Therefore,  we  have,  for  a  third  approximation  to  the  value  of  E, 
JE=119°43'44".64, 

which  requires  no  further  correction,  since  it  satisfies  the  equatior 
between  .If  and  E. 

To  find  r  and  v,  we  have 

1/r  sin  \v  =  I/a  (1  -f  e)  sin  £1?, 


T/r  cos^v  =  l/a(l  —  e)  cos  J.E, 

The  values  of  the  first  factors  in  the  second  members  of  these 
equations  are:  log  l/ofT+l)  =  0.2328104,  and  logVa(l—e}  = 
0.1468741;  and  we  obtain 

v  =  129°  3'  50".52,  log  r  =  0.4282854 

Since  n  —  £  =  197°  37'  49".58,  we  have 

w  =  v_|_^_^^  326°  41'  40".10. 

The  heliocentric  co-ordinates  in  reference  to  the  equator  as  the  fun- 
damental plane  are  then  derived  from  the  equations 

x  =  r  sin  a  sin  (  A  -f-  u), 
y  =r  sin  b  sin  (B  -f-  it), 
z  =  r  sin  c  sin  (  C  -f-  1*), 

which  give,  for  Eurynome, 

x  =  —  2.6611270,  y  =  +  0.3250277,  2  =  +  0.0119486. 

The  American  Nautical  Almanac  gives,  for  the  equatorial  co-ordi- 
nates of  the  sun  for  1865  February  24.5  mean  time  at  Washington, 
referred  to  the  mean  equinox  and  equator  of  the  beginning  of  the 
year, 

X=  +  0.9094557,  Y=  —  0.3599298,  Z=  —  0.1561751. 

Finally,  the  geocentric  right  ascension,  declination,  and  distance  are 
given  by  the  equations 


y-f  F 


= 


the   first  form  of  the  equation  for  tan£  being  used  when  sin  a  is 
greater  than  cos  a. 

The  value  of  A  must  always  be  positive;   and  8  cannot  exceed 
zfc  90°,  the  minus  sign  indicating  south  declination.   Thus,  we  obtain 


NUMEKICAL   EXAMPLES.  10  < 

a  =  181°  8'  29".29,        3  =  —  4°  42'  21".56,        log  J  =  0.2450054. 

To  reduce  a  and  8  to  the  true  equinox  and  equator  of  February 
24.5,  we  have,  from  the  Nautical  Almanac, 

/=  +  16".80,  log  0  =  1.0168,  £  =  45°  16'; 

and.  substituting  these  values  in  equations  (110),  the  result  is 
AO  =  +  17".42,  A<5  =  —  7".17. 

Hence  the  geocentric  place,  referred  to  the  true  equinox  and  equator 
of  the  date,  is 

a  =  181°  8'  46".71,  3  =  —  4°  42'  28".73,  log  A  =  0.2450054. 

When  only  a  single  place  is  required,  it  is  a  little  more  expeditious 
to  compute  r  from 

r  =  a  (1  —  e  cos  jE), 
and  then  v  —  E  from 

sin  i  (v  —  JE)  =  \-  sin  ±<p  sin  E. 
\r 

Thus,  in  the  case  of  the  required  place  of  Eurynome,  we  get 

log  r  =  0.4282852,  v  —  E  =  9°  20'  5".92, 

v  =  129°3'50".56, 

agreeing  with  the  values   previously  determined.     The   calculation 
may  be  proved  by  means  of  the  formula 

sin  ^  (v  -f-  E)  =  'y-  cos  4 <p  sin  E. 

In  the  case  of  the  values  just  found,  we  have 

J  (v  +  jE)  =  124°  23'  47".60,  log  sin  J  (v  +  E)  =  9.9165316. 

while  the  second  member  of  this  equation  gives 
log  sin  1  (v  +  E)  =  9.9165316 

In  the  calculation  of  a  single  place,  it  is  also  very  little  shorter  to 
compute  first  the  heliocentric  longitude  and  latitude  by  means  of  the 
equations  (82),  then  the  geocentric  latitude  and  longitude  by  means 
of  (89)  or  (90),  and  finally  convert  these  into  right  ascension  and 
declination  by  means  of  (92).  When  a  large  number  of  places  are 
to  be  computed,  it  is  often  advantageous  to  compute  the  heliocentrio 


108  THEORETICAL   ASTRONOMY. 

co-ordinates  directly  from  the  eccentric  anomaly  by  means  of  the 
equations  (105). 

The  calculation  of  the  geocentric  place  in  reference  to  the  ecliptic 
is,  in  all  respects,  similar  to  that  in  which  the  equator  is  taken  as  the 
fundamental  plane,  and  does  not  require  any  further  illustration. 

The  determination  of  the  geocentric  or  heliocentric  place  in  the 
cases  of  parabolic  and  hyperbolic  motion  differs  from  the  process 
indicated  in  the  preceding  example  only  in  the  calculation  of  r  and  v. 
To  illustrate  the  case  of  parabolic  motion,  let  t  —  T=  75.364  days; 
log 5  =  9.9650486;  and  let  it  be  required  to  find  r  and  v. 

First,  we  compute  m  from 

CL 


in  which  log  (70=  9.9601277,  and  the  result  is 

log  m  =  0.0125548. 
Then  we  find  M  from 

which  gives 

log  M=  1.8897187. 

From  this  value  of  log  M  we  derive,  by  means  of  Table  VI., 

v  =  79°  55'  57".26. 
Finally,  r  is  found  from 

cos2  jy 

which  gives 

log  r  =  0.1961120. 

For  the  case  of  hyperbolic  motion,  let  there  be  given  t  —  T= 
65.41236  days;  4>  =  37°  35'  0".0,  or  log 6  =  0.1010188;  and  lug  a 
=  0.6020600,  to  find  r  and  v.  First,  we  compute  ^Vfrom 

"  at 
in  which  log  A  =  9.6377843,  and  we  obtain 

logN=  8.7859356;  N=  0.06108514. 

The  value  of  F  must  now  be  found  from  the  equation 
N=  el  tan  F  —  log  tan  (45°  +  £  F). 


NUMERICAL   EXAMPLES.  109 

If  we  assume  F=  30°,  a  more  approximate  value  may  be  derived 
from 


which  gives  F,  =  28°  40'  23",  and  hence  N,  =  0.072678.  Then  we 
compute  the  correction  to  be  applied  to  this  value  of  F,  by  means  of 
the  equation 

(N-N,^oS*F 

'    l(e  —  cosF,-)    *' 
wherein  s  =  206264".8;  and  the  result  is 

*F,  =  4.6097  (N—  Nf~)  s  =  —  3°  3'  43".0. 
Hence,  for  a  second  approximation  to  the  value  of  F}  we  have 

.F,  =  25036'40".0. 

The  corresponding  value  of  Nis  N,  —  0.0617653,  and  hence 
AJF,  =  5.199  (N—  NJ  s  =  —  12'  9".4. 


The  third  approximation,  therefore,  gives  F,  =  25°  24'  30".6,  and, 
repeating  the  operation,  we  get 

JFT=25°24'27".74. 

which  requires  no  further  correction. 
To  find  r,  we  have 


which  gives 

log  r  =  0.2008544. 

Then,  v  is  derived  from 

tan  ±v  —  cot  ^4  tan  ±F, 
and  we  find 

v  =  67°  3'  0".0. 

When  several  places  are  required,  it  is  convenient  to  compute 
and  r  by  means  of  the  equations 


110  THEORETICAL   ASTRONOMY. 

For  the  given  values  of  a  and  e  we  have  log  V  a(e  +  1)  =  0.4782649, 
logl/a(e  —  1)  =  0.0100829,  and  hence  we  derive 

v  =  67°  2'  59".92,  log  r  =  0.2008545. 

It  remains  yet  to  illustrate  the  calculation  of  v  and  r  for  elliptic 
and  hyperbolic  orbits  in  which  the  eccentricity  differs  but  little  from 
unity.  First,  in  the  case  of  elliptic  motion,  let  t  —  T=  68.25  days; 
e  =  0.9675212;  and  log  q  =  9.7668134.  We  compute  M  from 


wherein  log  C0=  9.9601277,  which  gives 

log  3f=  2.1404550. 
\Yith  this  as  argument  we  get,  from  Table  VI., 

V=  101°  38'  3".74, 
and  then  with  this  value  of  V  as  argument  we  find,  from  Table  IX., 

A  --•=  1540".08,  B  =  9".506,  C=  0".062. 

^  _  e 

Then  we  have  log  i  =  log  —   —  =  8.217680,  and  from  the  equation 

1  -f-  e 

v  =  V+  -4(1000 


we  get 

v  =  V+  42'  22".28  -f  25".90  -f  0".28  =  102°  20'  52".20. 

The  value  of  r  is  then  found  from 


-f  e  cos  v' 
namely, 

log  r  =  0.1614051. 

We  may  also  determine  r  and  v  by  means  of  Table  X.     Thus,  we 
first  compute  M  from 


Assuming  B  =  1,  we  get  log  M=  2.13757,  and,  entering  Table  VI. 
with  this  as  argument,  we  find  w  =  101°  25'.  Then  we  compute  A 
from 

5(l--e 
A~  *   1       9e 


NUMERICAL   EXAMPLES.  Ill 

which  gives  A  =  0.024985.     With  this  value  of  A  as  argument,  we 
find,  from  Table  X., 

log  B  =  0.0000047. 

The  exact  value  of  M  is  then  found  to  be 

log  M=  2.1375635, 
which,  by  means  of  Table  VI.,  gives 

w  =  101°  24'  36".26. 
By  means  of  this  we  derive 

A  =  0.02497944, 
and  hence,  from  Table  X., 

log  C  =0.0043771. 
Then  we  have 

which  gives 

v  =  102°  20'  52".20, 

agreeing  exactly  with  the  value  already  found.   Finally,  r  is  given  by 




"  (1  -{-AC2)  cos1}*' 
from  which  we  get 

log  r  =  0.1614052. 

Before  the  time  of  perihelion  passage,  t  —  T  is  negative ;  but  the 
value  of  v  is  computed  as  if  this  were  positive,  and  is  then  considered 
as  negative. 

In  the  case  of  hyperbolic  motion,  i  is  negative,  and,  with  this  dis- 
tinction, the  process  when  Table  IX.  is  used  is  precisely  the  same 
as  for  elliptic  motion;  but  when  table  X.  is  used,  the  value  of  A 
must  be  found  from 

5(e—  1) 
^(f+ 
and  that  of  r  from 


(1— .AC2)  cos8  in' 

the  values  of  log  B  and  log  C  being  taken  from  the  columns  of  the 
table  which  belong  to  hyperbolic  motion. 

In  the  calculation  of  the  position  of  a  comet  in  space,  if  the  motion 


112  THEORETICAL   ASTRONOMY. 

is  retrograde  and  the  inclination  is  regarded  as  less  than  90°,  the  dis- 
tinctions indicated  in  the  formulae  must  be  carefully  noted. 

42.  When  we  have  thus  computed  the  places  of  a  planet  or  comet 
for  a  series  of  dates  equidistant,  we  may  readily  interpolate  the  places 
for  intermediate  dates  by  the  usual  formulae  for  interpolation.  The 
interval  between  the  dates  for  which  the  direct  computation  is  made 
should  also  be  small  enough  to  permit  us  to  neglect  the  effect  of  the 
fourth  differences  in  the  process  of  interpolation.  This,  however,  is 
not  absolutely  necessary,  provided  that  a  very  extended  series  of 
places  is  to  be  computed,  so  that  the  higher  orders  of  differences  may 
be  taken  into  account.  To  find  a  convenient  formula  for  this  inter- 
polation, let  us  denote  any  date,  or  argument  of  the  function,  by 
a  +  na>>  and  the  corresponding  value  of  the  co-ordinate,  or  of  the 
function,  for  which  the  interpolation  is  to  be  made,  by  /  (a  -f  nco). 
If  we  have  computed  the  values  of  the  function  for  the  dates,  or 
arguments,  a  —  to,  a,  a  -f  a>,  a  -f-  2w,  &c.,  we  may  assume  that  an 
expression  for  the  function  which  exactly  satisfies  these  values  will 
also  give  the  exact  values  corresponding  to  any  intermediate  value 
of  the  argument.  If  we  regard  n  as  variable,  we  mav  expand  the 
function  into  the  series 

f(a  +  no>)  =/(«)  +  An  +  JSn9  +  On9  -f  &c.  (116) 

and  if  we  regard  the  fourth  differences  as  vanishing,  it  is  only  neces- 
sary to  consider  terms  involving  ns  in  the  determination  of  the 
unknown  coefficients  A,  B,  and  C.  If  we  put  n  successively  equal 
to  —  1,  0,  1,  and  2,  and  then  take  the  successive  differences  of  these 
values,  we  get 

I.  Diff.  II.  Diff.     III.  Diff. 

/(<,-«)   =f(a)-A   +  B  -C 


f(a  +  2«>)  =/(o)  +  2A  +  4B  +  8  C 
If  we  symbolize,  generally,  the  difference  f(a  +  nco)  —  f(a  -\-(n  —  1  )  <o) 


(n—  |)  o>),  the  difference  /  (a  +  (n-+  J)  a>)—  /(a  +  (n—  J)  a») 
by  /"  (a  -f-  wa>),  and  similarly  for  the  successive  orders  of  differences^ 
these  may  be  arranged  as  follows  :  — 

Argument.  Function.  I.  Diff.  II.  Diff.  III.  Diff. 

a  —  a>  /(a_a/) 

a  /(a)  /(«-iK>        f  (a) 


INTERPOLATION.  113 

Comparing  these  expressions  for  the  differences  with  the  above,  we 
get 

C=J/"(a  +  i«),  J»  =  i/"(a), 

A  =f  (a  +  J»)  -  if'  (a)  -  J/»  (a  +  », 

which,  from  the  manner  in  which  the  differences  are  formed,  give 

C=  $  (/"  (a  4-  «,)  -/"  (a)),  B  =  if"  (a), 

A  =/  (a  +  «»)  -/(a)  -  i/"  (a)  -  J  (/"  (a  +  «,)  -/"  (a)). 

To  find  the  value  of  the  function  corresponding  to  the  argument 
a  +  Jw,  we  have  n  =  £,  and,  from  (116), 

/(a  +  £  «)  =/(a)  +  14  +  ^  +  4  a 

Substituting  in  this  the  values  of  A,  B,  and  (7,  last  found,  and  re- 
ducing, we  get 

/(a  -H«0  =  i  (/(a  +  «0  +/(«))  ~  I  (i  (/"  («  +  «)  +/"  W)), 

in  which  only  fourth  differences  are  neglected,  and,  since  the  place 
of  the  argument  for  n  =  0  is  arbitrary,  we  have,  therefore,  generally, 


/(a  +  (n  +  £)  «»)  =  i  (/(«  +  (*  +  !)  ")  +/(a  +  no,)) 

-  4  (i  (/'  («+(»  +  1)  «0  +/'  («  +  »«)))•  (1.17) 

Hence,  to  interpolate  the  value  of  the  function  corresponding  to  a 
date  midway  between  two  dates,  or  values  of  the  argument,  for  which 
the  values  are  known,  we  take  the  arithmetical  mean  of  these  two 
known  values,  and  from  this  we  subtract  one-eighth  of  the  arith- 
metical mean  of  the  second  differences  which  are  found  on  the  same 
horizontal  line  as  the  two  given  values  of  the  function. 

By  extending  the  analytical  process  here  indicated  so  as  to  include 
the  fourth  and  fifth  differences,  the  additional  term  to  be  added  to 
equation  (117)  is  found  to  be 

«  +  no,))), 


and  the  correction  corresponding  to  this  being  applied,  only  sixth 
differences  will  be  neglected. 

It  is  customary  in  the  case  of  the  comets  which  do  not  move  too 
rapidly,  to  adopt  an  interval  of  four  days,  and  in  the  case  of  the 
asteroid  planets,  either  four  or  eight  days,  between  the  dates  for  which 
the  direct  calculation  is  made.  Then,  by  interpolating,  in  the  case  of 
an  interval  co,  equal  to  four  days,  for  the  intermediate  dates,  we 
obtain  a  series  of  places  at  intervals  of  two  days ;  and,  finally,  inter- 

8 


114  THEORETICAL   ASTRONOMY. 

polating  for  the  dates  intermediate  to  these,  we  derive  the  places  at 
intervals  of  one  day.  When  a  series  of  places  has  been  computed, 
the  use  of  differences  will  serve  as  a  check  upon  the  accuracy  of  the 
calculation,  and  will  serve  to  detect  at  once  the  place  which  is  not 
correct,  when  any  discrepancy  is  apparent.  The  greatest  discordance 
will  be  shown  in  the  differences  on  the  same  horizontal  line  as  the 
erroneous  value  of  the  function  ;  and  the  discordance  will  be  greater 
and  greater  as  we  proceed  successively  to  take  higher  orders  of  dif- 
ferences. In  order  to  provide  against  the  contingency  of  systematic 
error,  duplicate  calculation  should  be  made  of  those  quantities  in 
which  such  an  error  is  likely  to  occur. 

The  ephemerides  of  the  planets,  to  be  used  for  the  comparison  of 
observations,  are  usually  computed  for  a  period  of  a  few  weeks  before 
and  after  the  time  of  opposition  to  the  sun  ;  and  the  time  of  the 
opposition  may  be  found  in  advance  of  the  calculation  of  the  entire 
ephemeris.  Thus,  we  find  first  the  date  for  which  the  mean  longitude 
of  the  planet  is  equal  to  the  longitude  of  the  sun  increased  by  180°  ; 
then  we  compute  the  equation  of  the  centre  at  this  time  by  means  of 
the  equation  (53),  using,  in  most  cases,  only  the  first  term  of  the 

development,  or 

v  —  M  = 


e  being  expressed  in  seconds.  Next,  regarding  this  value  as  con- 
stant, we  find  the  date  for  which 

L  -J-  equation  of  the  centre 

is  equal  to  the  longitude  of  the  sun  increased  by  180°  ;  and  for  this 
date,  and  also  for  another  at  an  interval  of  a  few  days,  we  compute 
u,  and  hence  the  heliocentric  longitudes  by  means  of  the  equation 

tan  (I  —  Q  )  =  tan  u  cos  i. 

Let  these  longitudes  be  denoted  by  I  and  lf,  the  times  to  which  they 
correspond  by  t  and  t'9  and  the  longitudes  of  the  sun  for  the  same 
times  by  O  and  Q  '  ;  then  for  the  time  t0,  for  which  the  heliocentric 
longitudes  of  the  planet  and  the  earth  are  the  same,  we  have 


or  (118) 


the  first  of  these  equations  being  used  when  I  —  180°  —  O  is  less 


TIME   OF   OPPOSITION.  115 

than  V  —  180°  —  O'.  If  the  time  t0  differs  considerably  from  t  or 
t',  it  may  be  necessary,  in  order  to  obtain  an  accurate  result,  to  repeat 
the  latter  part  of  the  calculation,  using  tQ  for  t.  and  taking  t'  at  a 
small  interval  from  this,  and  so  that  the  true  time  of  opposition  shall 
fall  between  t  and  t'  .  The  longitudes  of  the  planet  and  of  the  sun 
must  be  measured  from  the  same  equinox. 

When  the  eccentricity  is  considerable,  it  will  facilitate  the  calcula- 
tion to  use  two  terms  of  equation  (53)  in  finding  the  equation  of  the 
centre,  and,  if  e  is  expressed  in  seconds,  this  gives 

„  _  M  =  2e  sin  Jf  +  5  .  t.  sin  2M9 

4     S 

8  being  the  number  of  seconds  corresponding  to  a  length  of  arc  equal 
to  the  radius,  or  206264".8  ;  and  the  value  of  v  —  M  will  then  be 
expressed  in  seconds  of  arc.  In  all  cases  in  which  circular  arcs  are 
involved  in  an  equation,  great  care  must  be  taken,  in  the  numerical 
application,  in  reference  to  the  homogeneity  of  the  different  terms. 
If  the  arcs  are  expressed  by  an  abstract  number,  or  by  the  length  of 
arc  expressed  in  parts  of  the  radius  taken  as  the  unit,  to  express  them 
in  seconds  we  must  multiply  by  the  number  206264.8  ;  but  if  the 
arcs  are  expressed  in  seconds,  each  term  of  the  equation  must  contain 
only  one  concrete  factor,  the  other  concrete  factors,  if  there  be  any, 
being  reduced  to  abstract  numbers  by  dividing  each  by  s  the  number 
of  seconds  in  an  arc  equal  to  the  radius. 

43.  It  is  unnecessary  to  illustrate  further  the  numerical  application 
of  the  various  formulae  which  have  been  derived,  since  by  reference 
to  the  formulae  themselves  the  course  of  procedure  is  obvious.  It 
may  be  remarked,  however,  that  in  many  cases  in  which  auxiliary 
angles  have  been  introduced  so  as  to  render  the  equations  convenient 
for  logarithmic  calculation,  by  the  use  of  tables  which  determine  the 
logarithms  of  the  sum  or  difference  of  two  numbers  when  the  loga- 
rithms of  these  numbers  are  given,  the  calculation  is  abbreviated, 
and  is  often  even  more  accurately  performed  than  by  the  aid  of  the 
auxiliary  angles. 

The  logarithm  of  the  sum  of  two  numbers  may  be  found  by  meana 
of  the  tables  of  common  logarithms.  Thus,  we  have 


If  we  put 

log  tan  x  =  J  (log  b  —  log  a), 


116  THEORETICAL   ASTRONOMY. 

we  shall  have 

log  (a  -f  6)  =  log  a  —  2  log  cos  x, 
or 

log  (a  -f-  6)  =  log  b  —  2  log  sin  x. 

The  first  form  is  used  when  cos  x  is  greater  than  sin  x,  and  the  second 
form  when  cos  a;  is  less  than  sin  a;. 

It  should  also  be  observed  that  in  the  solution  of  equations  of  the 
form  of  (89),  after  tan  (A —  0) — using  the  notation  of  this  particular 
case — has  been  found  by  dividing  the  second  equation  by  the  first, 
the  second  members  of  these  equations  being  divided  by  cos  (A  —  0) 
and  sin  (^  —  0 ),  respectively,  give  two  values  of  J  cos  /?,  which  should 
agree  within  the  limits  of  the  unavoidable  errors  of  the  logarithmic 
tables ;  but,  in  order  that  the  errors  of  these  tables  shall  have  the 
least  influence,  the  value  derived  from  the  first  equation  is  to  be  pre- 
ferred when  cos  (A  —  ©)  is  greater  than  sin  (^  —  0),  and  that  derived 
from  the  second  equation  when  cos  (A  —  0)  is  less  than  sin  (^  —  ©). 
The  value  of  J,  if  the  greatest  accuracy  possible  is  required,  should 
be  derived  from  J  cos  /9  when  /9  is  less  than  45°,  and  from  J  sin  /? 
when  /?  is  greater  than  45°. 

In  the  application  of  numbers  to  equations  (109),  when  the  values 
of  the  second  members  have  been  computed,  we  first,  by  division, 
find  tan|(&'  -f-  ^0)  an(^  tanj(&' —  fti0);  then,  if  sin|(&' -f  w0)  is 
greater  than  cos|(&'  +  w0),  we  find  cos^'  from  the  first  equation; 
but  if  sin  f  (&>'  -f  <w0)  is  less  than  cos|  (&'  -f  <w0),  we  find  cos^v  from 
the  second  equation.  The  same  principle  is  applied  in  finding  sin  \ir 
by  means  of  the  third  and  fourth  equations.  Finally,  from  sin^' 
and  cos  \if  we  get  tan  %i' \  and  hence  if.  The  check  obtained  by  the 
agreement  of  the  values  of  sin  \V  and  cos  |i',  with  those  computed 
from  the  value  of  V  derived  from  tan  |zv,  does  not  absolutely  prove 
the  calculation.  This  proof,  however,  may  be  obtained  by  means  of 

the  equation 

sin  ir  sin  Q, '  =  sin  i  sin  Q, , 
or  by 

sin  i'  sin  o>0  =  sin  e  sin  & . 

In  all  cases,  care  should  be  taken  in  determining  the  quadrant  in 
which  the  angles  sought  are  situated,  the  criteria  for  which  are  fixed 
either  by  the  nature  of  the  problem  directly,  or  by  the  relation  of  the 
algebraic  signs  of  the  trigonometrical  functions  involved. 


DIFFERENTIAL   FORMULAE.  117 


CHAPTER  II. 

INVESTIGATION  OF  THE  DIFFERENTIAL  FORMULAE  WHICH  EXPRESS  THE  RELATION 
BETWEEN  THE  GEOCENTRIC  OR  HELIOCENTRIC  PLACES  OF  A  HEAVENLY  BODY 
AND  THE  VARIATION  OF  THE  ELEMENTS  OF  ITS  ORBIT. 

44.  IN  many  calculations  relating  to  the  motion  of  a  heavenly 
body,  it  becomes  necessary  to  determine  the  variations  which  small 
increments  applied  to  the  values  of  the  elements  of  its  orbit  will  pro- 
duce in  its  geocentric  or  heliocentric  place.  The  form,  however,  in 
which  the  problem  most  frequently  presents  itself  is  that  in  which 
approximate  elements  are  to  be  corrected  by  means  of  the  differences 
between  the  places  derived  from  computation  and  those  derived  from 
observation.  In  this  case  it  is  required  to  find  the  variations  of  the 
elements  such  that  they  will  cause  the  differences  between  calculation 
and  observation  to  vanish ;  and,  since  there  are  six  elements,  it  follows 
that  six  separate  equations,  involving  the  variations  of  the  elements 
as  the  unknown  quantities,  must  be  formed.  Each  longitude  or  right 
ascension,  and  each  latitude  or  declination,  derived  from  observation, 
will  furnish  one  equation ;  and  hence  at  least  three  complete  observa- 
tions will  be  required  for  the  solution  of  the  problem.  When  more 
than  three  observations  are  employed,  and  the  number  of  equations 
exceeds  the  number  of  unknown  quantities,  the  equations  of  condi- 
tion which  are  obtained  must  be  reduced  to  six  final  equations,  from 
which,  by  elimination,  the  corrections  to  be  applied  to  the  elements 
may  be  determined. 

If  we  suppose  the  corrections  which  must  be  applied  to  the  ele- 
ments, in  order  to  satisfy  the  data  furnished  by  observation,  to  be  so 
small  that  their  squares  and  higher  powers  may  be  neglected,  the 
variations  of  those  elements  which  involve  angular  measure  being 
expressed  in  parts  of  the  radius  as  unity,  the  relations  sought  may 
be  determined  by  differentiating  the  various  formulae  which  determine 
the  position  of  the  body.  Thus,  if  we  represent  by  6  any  co-ordi- 
nate of  the  place  of  the  body  computed  from  the  assumed  elements 
of  the  orbit,  we  shall  have,  in  the  case  of  an  elliptic  orbit, 


118  THEORETICAL   ASTRONOMY. 

MQ  being  the  mean  anomaly  at  the  epoch  T.  Let  6f  denote  the  value 
of  this  co-ordinate  as  derived  directly  or  indirectly  from  observation  ; 
then,  if  we  represent  the  variations  of  the  elements  by  ATI,  A&,  Ai, 
&c.,  and  if  we  suppose  these  variations  to  be  so  small  that  their 
squares  and  higher  powers  may  be  neglected,  we  shall  have 


do  do  ,    do  do 

d»A*+5aAa+-3rA<+27 

do       ....       dO 


„. 


The  differential  coefficients  -1—  ,  ^  —  ,  &c.  must  now  be  derived  from 

d*    a$> 
the  equations  which  determine  the  place  of  the  body  when  the  ele- 

ments are  known. 

We  shall  first  take  the  equator  as  the  plane  to  which  the  positions 
of  the  body  are  referred,  and  find  the  differential  coefficients  of  the 
geocentric  right  ascension  and  declination  with  respect  to  the  elements 
of  the  orbit,  these  elements  being  referred  to  the  ecliptic  as  the  fun- 
damental plane.  Let  x,  y,  z  be  the  heliocentric  co-ordinates  of  the 
body  in  reference  to  the  equator,  and  we  have 

9  =/(*,  y,  z), 
or 

do  .     ,    do  ,     ,    do  , 
dO  =  -j-  dx  -4-  -j-  dy  4-  -,-  dz. 
dx         r  dy    y       dz 

Hence  we  obtain 

dO  _  d0_    dx        dO_    dy_      dO_    dz  ; 
~dn        dx  '  d-K        dy'dTtdz'dn* 

and  similarly  for  the  differential  coefficients  of  0  with  respect  to  the 
other  elements.  We  must,  therefore,  find  the  partial  differential  co- 
efficients of  d  with  respect  to  #,  y,  and  z,  and  then  the  partial  differen- 
tial coefficients  of  these  co-ordinates  with  respect  to  the  elements.  In 
the  case  of  the  right  ascension  we  put  0  =  oc,  and  in  the  case  of  the 
declination  we  put  6  =  3. 

45.  If  we  differentiate  the  equations 

x  +  X=  A  cos  8  cos  a, 
y  -{-  Y=  A  cos  d  sin  a, 
z  -\-  Z  =  A  sin  d, 

regarding  X,  F,  and  Z  as  constant,  we  find 


DIFFERENTIAL   FORMULAE.  119 

dx  =  cos  u  cos  8  d 4  —  A  sin  a  cos  8  da,  —  A  cos  a  sin  d  dd, 
dy  ==  sin  a  cos  d  d  A  -\-  A  cos  a  cos  $  da,  —  A  sin  a  sin  d  dd, 
dz  =  sin  d  d  A  -j-  A  cos  d  dd. 

From  these  equations,  by  elimination,  we  obtain 

..   ,             sin  a  ,        cos  a  ,  /rk>l 

cos  d  da,  = —  az  -| -p  cfy,  (3) 

,,  cos  a  sin  d  sin  a  sin  d  cos<5 

<**  =       — — dx~    — 


Therefore,  the  partial  differential  coefficients  of  a  and  d  with  respect 
to  the  heliocentric  co-ordinates  are 

da,  sin  a  dS  cos  a  sin  <5 

dx  ~  A  '  dx  ~  A 

„  do,       cos  a  a*<5  sin  a  sin  5  ... 

cos  #  -j-  =  — -j— ,  T-  = -. ,  (4) 

dy          A  dy  A 

da.  dd       cos# 

—j—  =•  0.  — j—  == ^ — • 

dz  dz          A 

Next,  to  find  the  partial  differential  coefficients  of  the  co-ordinates 
x,  y,  Zj  with  respect  to  the  elements,  if  we  differentiate  the  equations 
(100)!,  observing  that  sin  a,  sin  6,  sin  c,  J.,  B,  C,  are  functions  of  ft 
and  iy  we  get 

dx  =  -  dr  -f  #  cot  ( J.  -|-  w)  efot  -J-  -j^-  rfft  -f  -^r  c^i, 


df2  =  ?  dr  +  2  cot  (0+  u)  du  +  ^  rf^  +  -jjjr  di. 

To  find  the  expressions  for  ~,  -^,  &c.,  we  have  the  equations 

d/oo    di' 

x  =  r  cos  w  cos  &  —  r  sin  M  sin  &  cos  i, 

y  —  r  cos  -w  sin  &  cos  e  -j-  r  sin  w  cos  ^  cos  i  cos  e  —  r  sin  u  sin  ^  sin  e, 

3  =  r  cos  u  sin  &  sin  s  -j-  r  sin  u  cos  &  cos  i  sin  e  -f  r  sin  u  sin  i  cos  e. 

which  give,  by  differentiation, 

-=^-  =  —  r  cos  u  sin  &  —  r  sin  u  cos  &  cos  i, 
d& 

_ J^  =  r  cos  w  cos  &  cos  e  —  r  sin  w  sin  ft  cos  i  cos  e, 


120  THEORETICAL   ASTRONOMY. 

dz 

-—  =  r  cos  u  cos  &  sin  e  —  r  sin  u  sin  &  cos  i  sin  e. 
«& 

dte 

-—  =  r  sm  it  sin  $6  sin  *> 
ew 

<fy 

—~  =  —  r  sm  u  cos  &  sm  ?  cos  s  —  r  sin  u  cos  t  sin  e, 

dl 

ds 
—p-  =  —  T  sm  it  cos  &  sm  i  sm  e  -f~  r  sin  it  cos  i  cos  e. 

The  first  three  of  these  equations  immediately  reduce  to 

dx  .  dy  dz  .          , 

j—  =  —  y  cos  e  —  z  sm  e,         -j~  —  x  cos  e,         -JQ  =  a;  sin  e  ;    (5) 

and  since 

cos  a  =  sin  £2  sin  t, 

cos  b  =  —  cos  &  sin  i  cos  e  —  cos  i  sin  e, 

cos  c  =  —  cos  &  sin  i  sin  e  -f-  cos  i  cos  e, 

we  have,  also, 

dx  dy  .  ,  dz 

——•  =  r  sm  u  cos  a,  —  £-  =  r  sm  it  cos  0.  —  vr-  =  r  sm  u  cos  c. 

efo  di  di 

Further,  we  have 

du  =  dv  +  dn  —  d&, 
and  hence,  finally, 

At 

efo  =  _  dr  -f-  a;  cot  (  J.  -|-  w)  c?v  +  a;  cot  (  A  -f-  w)  ctr 


4-  (  —  x  cot  (  A  -f-  w)  —  y  cose  —  2  sin  e)  c?&  +  r  sin  ^  cos  a  di, 
cot  (5  +  u)  dv  +  2/  co 


—  /  cot  (P  +  w)  +  x  cos  e)  ^^  +  r  sin  tt  cos  b  di, 


dz  =  -dr-i-z  cot(0+  u)dv  +  z  cot(0+  u)  dn 
+  (—  z  cot(C+tO  +  x  sine)d&  +  r  smu  cose  di. 

These  equations  give,  for  the  partial  differential  coefficients  of  the 
heliocentric  co-ordinates  with  respect  to  the  elements, 


dz        dz 


DIFFERENTIAL  FORMULJ3.  121 

-jj-  =  —  x  cot  (A+u)—  y  cos  e—  z  sin  e,         ^-j-=  —  y  cot  (JB+t*)-f-a;  cos  e, 
JQ  =  —  z  cot(C+u)  +  x  sine; 

dx  dy  dz 

-JT-  —  r  sm  u  cos  a,          ~Ji=r  sm  w  cos  ^»          ~dr==r  sin  w  cos  c;  ^ 

da;  _  a;  dy  _  y  ds  _  z 

dr       r*  c?r       F  "3r"      f" 

When  the  direct  inclination  is  greater  than  90°,  if  we  introduce  the 
distinction  of  retrograde  motion,  we  have 

du  =  dv  —  dit  -f-  dQ, 
and  hence 


rfa?       da;  .  dy       dy    .  dz       dz    . 

-j—-=-j  --  ycose  —  2  sine,  j^=~j  +*CO§*,  j—  -=-5—  +  a;sme. 
o^  dv  d&  dv  d^  dv 

The  expressions  for  -=—  »  -^->  and  —  remain  unchanged;  and  we 
have,  also, 

da;  dy  .       dz  ,rtN 

—T^-  =  —  rsmttcosa.  -^n-  =  —  rsmwcoso.  -77-  =  —  rsmttcosc.  (9) 
di  di  di 

It  is  advisable,  in  order  to  avoid  the  use  of  two  sets  of  formulae,  in 
part,  to  regard  the  motion  as  direct  and  the  inclination  as  susceptible 
of  any  value  from  0°  to  180°.  If  the  elements  which  are  given  are 
for  retrograde  motion,  we  take  the  supplement  of  i  instead  of  i;  and 
if  we  designate  the  longitude  of  the  perihelion,  when  the  motion  is 
considered  as  being  retrograde,  by  (TT),  we  shall  have 


If  we  introduce,  as  one  of  the  elements  of  the  orbit,  the  distance 
of  the  perihelion  from  the  ascending  node,  we  have 

du  =  dv  -f~  d*0* 
and,  hence, 

dx        dx 


122  THEORETICAL   ASTKONOMY. 

The  values  of  —  >  -—  ,  and  —  must,  in  this  case,  be  found  by  means 

of  the  equations  (5). 

By  means  of  these  expressions  for  the  differential  coefficients  of  the 
co-ordinates  x,  y,  z,  with  respect  to  the  various  elements,  and  those 
given  by  (4),  we  may  derive  the  differential  coefficients  of  the  geo- 
centric right  ascension  and  declination  with  respect  to  the  elements 
&,  ij  and  n  or  o>,  and  also  with  respect  to  r  and  v,  by  writing  suc- 
cessively a  and  d  in  place  of  6,  and  &,  i,  &c.,  in  place  of  TT  in  the 
equation  (2).  The  quantities  r  and  v,  however,  are  functions  of  the 
remaining  elements  y>,  MQ,  and  /*;  and  we  have 

,         dr    ,  dr     ,,..        dr   , 


,         dv  j          dv    ,  f        dv   , 
dv  =^  -j-  d(p  -\-  -J1rF  dMQ  -f-  -j—  dp. 
d<p  dM0  d/j. 

Therefore,  the  partial  differential  coefficients  of  x,  with  respect  to 
the  elements  <f>,  Jf0,  and  //,  are 

dx 
d<p 
dx 


dx 

~dr" 
dx 
dr  ' 
dx 
dr  ' 

dr 

dr 
dM0 
dr 
dp 

+ 
+ 

+ 

dx 

dv 
dx 
~dv~ 
dx 
dv 

dv 

dv 
'dM^' 
dv 
'  dfi' 

dx 


The  expressions  for  the  partial  differential  coefficients  in  the  case  of 
the  co-ordinates  y  and  z  are  of  precisely  the  same  form,  and  are  ob- 
tained by  writing,  successively,  y  and  z  in  place  of  x.  The  values  of 

dx     dx     dy     dy     dz         ,   dz  .         1,1  ..         /^\ 

--—   -—  ,  —£-,  —  —,  ——,  and  —r-  are  given  by  the  equations  (7),  and 

dr     dv     dr     dv     dr  dv 

„      dr     dv      dr       dv      dr         ,   dv  , 

when  the  expressions  for  -7-,  —r-  »  —  —,  -j^pi  -r~>  and  —  have  been 

aM     d/j.  a/j. 


found,  the  partial  differential  coefficients  of  the  heliocentric  co-ordj 
nates  with  respect  to  the  elements  tp,  Mw  and  //  will  be  completely 
determined,   and   hence,    by   means  of  (2),    making   the   necessary 
changes,  the  differential  coefficients  of  a  and  d  with  respect  to  these 
elements. 

46.  If  we  differentiate  the  equation 
M=E— 


DIFFERENTIAL   FORMULA.  123 

we  shall  have 

dM=  dE(l  —  e  cos  E)  —  cos  <p  sin  E  d<p. 

But,  since  1  —  e  cos  E  =  -,  and  cos  <p  smE=-  sin  v,  this  reduces  to 

a  a 


or 


V  T 

dM=  -  dE  —  -  sin  v  dy, 
a  a 


dE  —  -  dM  -f  sin  v  dy. 


If  we  take  the  logarithms  of  both  members  of  the  equation 

tan  ±v  =  tan  ^tan  (45°  -f  £p), 
and  differentiate,  we  find 

dv  dE 


'        O    nJ, 


2  sin  £v  cos  £v       2  sin  ^  cos  ^  '   2  sin  (45°  +  J?)  cos  (45°  -f 
which  reduces  to 


snv   ,          snv 


Introducing  into  this  equation  the  value  of  dE,  already  found,  and 


..-.  ,      r  sin  v 

replacing  sin  E  by  -  >  we  get 
J  a  cosy 


a?  cos  y  j  ,  ,  .    sin  v  / 

dv  =  -  —  -  --- 


sn  v    a  cos 


2  ^       .  \  , 
-  +  \\d<p. 
1 


r2  cos^\       r 

But  since  a,  cos2^>  =PJ  and  -  =  1  +  sin  <f>  cos  v,  this  becomes 

QJ      COS  (P  ^  \      •  •?  /••<  r*N 

dv  = dM  -4- 1 h  tan  y  cos  v]  sm  v  c?^.  (12) 

r2 


If  we  differentiate  the  equation 

r  =  a  (1  —  e  co»  .E), 
we  shall  have 

dr  =  -  da  -f  ae  sin  .E  <iE  —  a  cos  ^  cos  jEJ  dy ; 
a 

and  substituting  for  dE  its  value  in  terms  of  dM  and  cfy>,  the  result 


is 


.__rl  ^a  _}-  a  tan  ^  sin-y  dM-\-  (ae  sin  ^J  sin  v  —  a  cos  p  cos  E)  dy. 


124  THEORETICAL  ASTRONOMY. 

XT          .          •     -n      sin  v  cos  <p  cos  v  4-  e 

Aow,  since  sin  .#==  —  -  —    —  ,  and  cos  E=  —  -  •  —  ,  we  shall  have 
1  -f  e  cos  v  1  -j-  e  cos  v 

.    r,  .  -r,      aecostfsin'v       a  cose?  (cos  V  -he) 

ae  sin  U  Bin  v  —  a  cos  f  cos  -E  =  -—  —  -  ---  =-—  -  —  -» 

1  +  e  cos  v  1  -j-  6  cos  v 

which  reduces  to 

ae  sin  E  sin  v  —  a  cos  <p  cos  ^  =  —  a  cos  p  cos  v 
Hence,  the  expression  for  dr  becomes 

dr  =  -  da  -j-  a  tan  ?>  sin  v  dM  —  a  cos  <p  cos  v  rf^>.  (14) 

Further,  we  have 


T  being  the  epoch  for  which  the  mean  anomaly  is  Mw  and 

A;  1/1  +  m 
^=  -  3  -- 
a^ 

Differentiating  these  expressions,  we  get 

<Of  =  dM0  +  (t  —  T}  dfJL, 
da  _        2     dfJL 
a~~      -*"7J 

and  substituting  these  values  in  the  expressions  for  dr  and  dv,  we 
have,  finally, 

(2r  i 
a  tan  p  sin  v  (t  —  I7)  —  —  J  dp 

—  a  cos  <?  cos  v  d<p,  (15) 


^^  _^        a        og     s.nv 

f*  rz  \  cos  <p  I 

From  these  equations  for  dr  and  dv  we  obtain  the  following  valuer 
of  the  partial  differential  coefficients  :  — 


dv      I    2  \  . 

=  —  a  cos  <p  cos  v,  —  -  =1  --  \-  tan  y  cos  v  Ism  v, 

^     \cos9?  / 

/MrtN 


dr 

=  —  a  cos  <p  cos  v,  —  -  = 

cos9? 

dr  dv 

_=atan?S1n»,  .    -^=  , 


DIFFEKENTIAL,   FORMULA.  125 

It  will  be  observed  that  in  the  last  term  of  the  expression  for  —  we 

have  supposed  fj.  to  be  expressed  in  seconds  of  arc,  and  hence  the 
factor  206264.8  is  introduced  in  order  to  render  the  equation  homo- 
geneous. 

47.  The  formulae  already  derived  are  sufficient  to  find  the  varia- 
tions of  the  right  ascension  and  declination  corresponding  to  the 
variations  of  the  elements  in  the  case  of  the  elliptic  orbit  of  a  planet; 
but  in  the  case  of  ellipses  of  great  eccentricity,  and  also  in  the  cases 
of  parabolic  and  hyperbolic  motion,  these  formulae  for  the  differential 
coefficients  require  some  modification,  which  we  now  proceed  to 
develop. 

First,  then,  in  the  case  of  parabolic  motion,  sin  <p  =  1,  and  instead 
of  MQ  and  fi  we  shall  introduce  the  elements  T  and  q,  the  differential 
coefficients  relating  to  TT,  Q,  ,  and  i  remaining  unchanged  from  their 
form  as  already  derived. 

If  we  differentiate  the  equation 


tan8 


regarding  T,  q,  and  v  as  variable,  we  shall  have 
kdT      _&£—  T    ,        ,  - 

q  +  i5 


or,  since  r2  =  <f  sec4  \v, 


Jf)  y.2 

•=—  d<l  +  2   "I  ^V« 


1/2  31/2 


Multiplying  through  by  -^-»  and  reducing,  we  get 


(17) 


Instead  of  q,  we  may  use  logg,  and  the  equation  will,  therefore, 
become 


in  which  y(0  is  the  modulus  of  the  system  of  logarithms. 


126  THEORETICAL   ASTRONOMY. 

If  we  take  the  logarithms  of  both  members  of  the  equation 

and  differentiate,  we  find 

dr  =  -  dq  -f-  r  tan  ^v  dv. 

Introducing  into  this  equation  the  value  of  dv  from  (17),  we  get 

^dT.        (19) 


q  r2? 

left  _  /p) 
Now,  since  -  ^    _     =  q  (tan  \v  -f  £tan3  Jt>),  and  q  =  r  cos2  Jv,  we  have 


1       3&  (*  —  T)  tan  Jv       1  0-21  •  «  i 

xi_  —  —  =-(14-  tan2  \v  —  3  sm2  \v  —  sm2  \v  tan2 
q  r*V  2q  r 

__  cosv 

r 
We  also  have 


tan  4v  — 


Therefore,  equation  (19)  reduces  to 

(20) 


,0^N 
(21) 


-. 

I/  2q 

If  we  introduce  cZ  log  q  instead  of  dg,  this  equation  becomes 


From  the  equations  (17),  (18),  (20),  and  (21),  we  derive 

dr  ksinv  dv  IcV 2q 

dT.  ~  V5q  5?  r2 

dr  dv  3&  (t  —  T) 


V^V/O    «/.  7  „  y >  V^^y 

dq  dq 


dr_    _qcosv  dv  3^  (t  —  T}  1/2? 

"  "  " 


and  then  we  have,  for  the  differential  coefficients  of  x  with  respect  to 
T  and  q  or  log  7, 


DIFFEEENTIAL   FOKMUL^.  127 

dx  _  dx     dr       dx     dv  dx       dx     dr       dx     dv 

dT==~d^"dT  +  ~d^"dT'  ~dq  =  ~dr  '  ~efy  +  ~dv  "dq 

dx          dx        dr          dx        dv 
' 


d  log  q       dr    d  log  q       dv  '  d  log  q' 

and  similarly  for  the  differential  coefficients  of  y  and  z  with  respect 
to  these  elements.  The  expressions  for  the  partial  differential  co- 
efficients of  x,  y,  and  z,  respectively,  with  respect  to  r  and  v  are  the 
same  as  already  found  in  the  case  of  elliptic  motion.  We  shall  thus 
obtain  the  equations  which  express  the  relation  between  the  variations 
of  the  geocentric  places  of  a  comet  and  the  variation  of  the  parabolic 
elements  of  its  orbit,  and  which  may  be  employed  either  to  correct 
the  approximate  elements  by  means  of  equations  of  condition  fur- 
nished by  comparison  of  the  computed  place  with  the  observed  place, 
or  to  determine  the  change  in  the  geocentric  right  ascension  and 
declination  corresponding  to  given  increments  assigned  to  the  ele- 
ments. 

48.  We  may  also,  in  the  case  of  an  elliptic  orbit,  introduce  T,  q, 
and  e  instead  of  the  elements  <p,  Mw  and  //.     If  we  differentiate  the 

expression 

q  =  a  (1  —  e\ 
we  shall  have 


2 


a  ,     ,   a   , 

da  =  -  da  +  —  de. 
q    •    r  q 

We  have,  also, 


in  which  Tis  the  time  of  perihelion  passage,  and 


dM  =  —  Jcl/T+^i  a-f  dT—  f  yH/F+~™  <*"*(*  —  ^)  da. 
Hence  we  derive 

dM=-  kVl^i  or^dT-l  (t  -  D  <tq 


Substituting  this  value  of  dM  in  equation  (12),  replacing  sin  <p  by  e, 
and  reducing,  we  get 


dv  =  — 


qr 


_  p  _(|  +  1  )  dn  .  )  j-L-  de.        (23) 


128  THEORETICAL   ASTJRONOMTY. 

In  a  similar  manner,  by  substituting  the  values  of  da  and  dM  in 
equation  (14),  and  reducing,  we  find 


&11  4-m 
dr  =  --  =!  -  e  sm  v  dT 


,  ir         kl+m(t—  T)         2         .      \ 

•f    -  —  |  -  '   .-  \  -  -  \hr—,  —  e  sm  v  I 
\S  1/2  g*  ^l  +  e  / 


.    (24) 


These  equations,  (23)  and  (24),  will  furnish  the  expressions  for  the 

*•  i   ^-.0?        4.-  i         cc.  •          dv    dv    dv    dr    dr  dr 

partial  differential  coefficients  -r—  ,  —  ,  —  ,  —  ,  —  ,  and  —  ,  which  are 

dT  dq    de    dT  dq  de 

required  in  finding  the  differential  coefficients  of  the  heliocentric  co- 
ordinates with  respect  to  the  elements  T,  g,  and  e,  these  quantities 
being  substituted  for  Jf0,  //,  and  <p,  respectively,  in  the  equations  (11). 

49.  When  the  orbit  is  a  hyperbola,  we  introduce,  in  place  of  Mw 
fjt,  and  <p,  the  elements  T,  q,  and  $. 
If  we  differentiate  the  equation 

N0  =  e  tan  F  —  loge  tan  (45°  + 
we  shall  have 


cos  F 
which  is  easily  transformed  into 

dF  „  tan  4/ 

+  tan  F 


^ 

a    cos  F  cos  ^ 

or 

dF  a  a    tan  4 


sin  F       r  tan  £  r     cos  4/ 

Let  us  now  take  the  logarithms  of  both  members  of  the  equation 

tan  %F  =  tan  Jt;  tan  ^4/, 
and  differentiate,  and  we  shall  have 

7  dF          sinv    , 

dv  =  sin  v  —. — ^ : d$. 

sin  jt<         sm  4< 

Introducing  into  this  equation  the  value  of  ^^  already  found,  we 
get 

a  sin  v   7  >T      /  a  sin  v    tan  4         sin  v  \  , 

ai?  = ^,dJ\n  — I 1 —  • h  — : I  «4'» 

r  tan  F  \     r         cos  ^         sin  ^  / 


DIFFERENTIAL   FORMULA.  129 

Bufc,  since  r  sin  v  =  a  tan  $  tan  F,  and  p  —  a  tan2  ^  this  reduces  to 

*  =  ^v5«i-(|  +  1)^d*.  (25) 

If  we  differentiate  the  equation 


we  get 

r  ,  ,  T,    d-F7  a        tan  4*   T 

dr  =  -  da  -f  ae  tan*  F—  —  ^  H  --  ^  --  —  cfy. 
a  smF       cosF     cos4- 

Trrr 

Substituting  in  this  equation  the  value  of  —  —  ^  we  obtain 

7        r  ,     ,   a?e  tan  F  7  .  _       /  a2e  tan2  jP          a     \  tan  4* 
dr  =  -  da  -\  --  dNtt  —  I  ----  =f  ) 

a  r  \        r  cos  jP/  cos  4 

which  is  easily  reduced  to 

r  7  sinv    7,T    .  pi     r  ae 

dr  =  ada+a^dN°  +  -r(^-^ 

But,  since 

r  ae  a 

cos  F  ~~  cos2  F  ~~  cos  F' 
this  reduces  to 

r 


or 

7        r  ,  sin  v    ,  ,r    .       cos  v    ,  ,0  n 

<Zr  =  _  ^a  +  a  -;  -  cZJVi  -f  p  —  -  a*.  (2o) 

a  r 


Now,  since  q  =  a(e  —  1),  we  have 

7        q  ,     .   a  tan  4/ 

da  =  -  da  -\ 

*       a  cos  4< 

or 

a 


, 
da  —  -dq . 

(/  ^  COS  ^ 

We  have,  also, 

tfffl_rfcrt(*—  21), 
and  hence 

^  =  _  y^a-  f  ^^_  pa-f  (^ 

By  substituting  the  value  of  c?a,  this  becomes 


130  THEORETICAL   ASTRONOMY. 

Substituting  this  value  of  dN0  in  equation  (25),  and  reducing,  we 
obtain 


qrz 

(27) 


qr* 

In  a  similar  manner,  substituting  in  equation  (26)  the  values  of 
da  and  dN0,  and  reducing,  we  get 


1/J9      cos  4  \S  1/2  g*       cos^T/cos-* 

l^kVp(t—  T)     sinv        /r  \    \  d* 

-f-    3  —  —  -  -  •  —   --   -  —  cos  V   p  1  -  —  •  (28) 

r\  q  cos^       \  r  /sin4 


The  equations  (27)  and  (28)  will  furnish  the  expressions  for  the 
partial  differential  coefficients  of  r  and  v  with  respect  to  the  elements 
T,  g,  and  <$/,  required  in  forming  the  equations  for  cos  d  da  and  dd. 
It  will  be  observed  that  these  equations  are  analogous  to  the  equa- 
tions (23)  and  (24),  and  that  by  introducing  the  relation  between  e 
and  ^  and  neglecting  the  mass,  they  become  identical  with  them. 
We  might,  indeed,  have  derived  the  equations  (27)  and  (28)  directly 
from  (23)  and  (24)  by  substituting  for  e  its  value  in  terms  of  a//;  but 
the  differential  formulae  which  have  resulted  in  deriving  them  directly 
from  the  equations  for  hyperbolic  motion,  will  not  be  superfluous. 

50.  It  is  evident,  from  an  inspection  of  the  terms  of  equations  (23), 
(24),  (27),  and  (28)  which  contain  de  and  d^,  that  when  the  value  of 
e  is  very  nearly  equal  to  unity,  the  coefficients  for  these  differentials 
become  indeterminate.  It  becomes  necessary,  therefore,  to  develop 
the  corresponding  expressions  for  the  case  in  which  these  equations 
are  insufficient.  For  this  purpose,  let  us  resume  the  equation 


2?* 
in  which  u  =  tan  Jw,  and  i  =  z  --  •     Then,  since 


=  l  +  K1  - 


we  shall  have 


DIFFERENTIAL   FORMULA.  131 


373it8  +  /gw7)  (1  —  e)2  -f-  &c.  (29) 

If  it  is  required  to  find  the  expression  for  -y-  in  the  case  of  the 

variation  of  the  elements  of  parabolic  motion,  or  when  1  —  e  is  very 
small,  we  may  regard  the  coefficient  of  1  —  e  as  constant,  and  neglect 
terms  multiplied  by  the  square  and  higher  powers  of  1  —  e.  By 
differentiating  the  equation  (29)  according  to  these  conditions,  and 
regarding  u  and  e  as  variable,  we  get 

0  =  (1  -f-  w2)  du  —  (\u  —  \u*  —  J-w5)  de; 
and,  since  du  =  J(l  +  u2)  dv,  this  gives 

$JL  =  &M--K—K  (30) 

The  values  of  the  second  member,  corresponding  to  different  values 
of  v,  may  be  tabulated  with  the  argument  v;  but  a  table  of  this  kind 

is  by  no  means  indispensable,  since  the  expression  for  -y-  may  be 

changed  to  another  form  which  furnishes  a  direct  solution  with  the 
same  facility.  Thus,  by  division,  we  have 

de  =  -l«  +  A 

and  since,  in  the  case  of  parabolic  motion, 


this  becomes 

*=A*C<=*2v^-it«.4*  -    (3!) 

If  we  differentiate  the  equation 


+  e  cos  v 
regarding  r,  v,  and  e  as  variables,  we  shall  have 

dr  _  2r2  sin2  \v   .    r*esinv     dv_ 
"  " 


132  THEORETICAL  ASTRONOMY. 

In  the  case  of  parabolic  motion,  e  =  1,  and  this  equation  is  easily 
transformed  into 

(33) 


Substituting  for  -j-  its  value  from  (31),  and  reducing,  we  get 


-       -     1/2, 

The  equations  (31)  and  (34)  furnish  the  values  of  —  and  —  to  be 

used  in  forming  the  expressions  for  the  variation  of  the  place  of  the 
body  when  the  parabolic  eccentricity  is  changed  to  the  value  1  -f-  de. 

When  the  eccentricity  to  which  the  increment  is  assigned  differs  but 

di) 
little  from  unity,  we  may  compute  the  value  of  —  directly  from 

equation  (30).  A  still  closer  approximation  would  be  obtained  by 
using  an  additional  term  of  (29)  in  finding  the  expression  for  —  /  but 

(JL& 

a  more  convenient  formula  may  be  derived,  of  which  the  numerical 
application  is  facilitated  by  the  use  of  Table  IX.  Thus,  if  we  differ- 
entiate the  equation 

v  =  v+  A  (1000  +  B  (lOOi)2  +  C(100i)8, 

regarding  the  coefficients  A,  jB,  and  C  as  constant,  and  introducing 
the  value  of  i  in  terms  of  e,  we  have 

J  =  ^_.|^___^i(10(H-)-4-^i(1000')    ' 

in  which  s  —  206264.8,  the  values  of  A,  B,  and  C9  as  derived  from 
the  table,  being  expressed  in  seconds.  To  find  — ,  we  have  „ 


Jc(t  — 


which  gives,  by  differentiation, 

Tc(t—T}         de  dV 


and  if  we  introduce  the  expression  for  the  value  of  M  used  as  the 
argument  in  finding  Fby  means  of  Table  VI.,  the  result  is 


DIFFERENTIAL   FORMULA.  133 

dV 

de 
Hence  we  have 


dv     JfcosF        200J.  400£  600(7    ,       ,f 

3~;(r+^~Kr^  (35) 


by  means  of  which  the  value  of  —  is  readily  found. 

de 

When  the  eccentricity  differs  so  much  from  that  of  the  parabola 

that  the  terms  of  the  last  equation  are  not  sufficiently  convergent, 

dv 
the  expression  for  —  ,  which  will  furnish  the  required  accuracy,  may 

be  derived  from  the  equations  (75)!  and  (76)r  If  we  differentiate  the 
first  of  these  equations  with  respect  to  e,  since  B  may  evidently  be 
regarded  as  constant,  we  get 

dw__    fl  k(t  —  T)  cosmic  (     . 

~  '  ' 


If  we  take  the  logarithms  of  both  members  of  equation  (76)j,  and 
differentiate,  we  get 

dv     _  dC  .     dw  __  4de  _ 
—  ~7T    i    „•    „,.       7i    i    r\~7i    i    n^" 


sinv        G       sinw 

To  find  the  differential  coefficient  of  C  with  respect  to  e,  it  will  be 
sufficient  to  take 


which  gives 
The  equation 
gives 


tan 
and  hence  we  obtain 


-  smw 
Substituting  this  value  in  equation  (37),  we  get 

dv  20  C3  C*sinv    dw  4smv 

sm  v  tan2  ^w  - 


134  THEOKETICAL   ASTRONOMY. 

and  substituting,  finally,  the  value  of  -=-,  we  obtain 

dv_         lc(t—T}  C2sinv          cos1  Aw         20  C> 

'ia^-Si: 
4sinw 


which,  by  means  of  (76)^  reduces  to 
dv  lct—T 


'  ' 


If  we  introduce  the  quantity  M  which  is  used  as  the  argument  in 
finding  w  by  means  of  Table  VI.,  this  equation  becomes 

dv  9  M  cos2  ^w  „„  .  8  tan  Av  f 

\6y) 


de  ~  2  (1  +  9e)  "  75  tan^w  v  (1  +  e)  (1  +  9e)' 

This  equation  remains  unchanged  in  the  case  of  hyperbolic  motion, 
the  value  of  C  being  taken  from  the  column  of  the  table  which  cor- 

responds to  this  case-;  and  it  will  furnish  the  correct  value  of  -=-  in 

de 

all  cases  in  which  the  last  term  of  equation  (23)  is  not  conveniently 

C?7* 

applicable.     The  value  of  --  is  then  given  by  the  equation  (32). 

When  the  eccentricity  differs  very  little  from  unity,  we  may  put 

B  =  1,  and  _ 

tan  \w  =  tan  £v  j/^  (i  +  ge), 

cos2  ^w  =  O2  cos2  %v. 
Then  we  shall  have 


^2  2k  (t—  T) 

C2  sm  v  —  —  .-   ,      cos4  4w. 
;  1/2  g* 

The  equation 


=  (1  +  A  C")  cos2  Jw  =  (1  +  i-^)  cos2>, 
gives 

|l  =  (1  +  p)  cos4  Jw  =  Ccos4  £w. 
Hence  we  derive 


—  T) 


NUMERICAL   EXAMPLES.  135 

[f  we  substitute  this  value  in  equation  (39),  and  put  C2  (1  +  e)  —  2, 
we  get 

im 


r2  (1  +  e)  (1  +  9e)' 

and  when  e  —  1,  this  becomes  identical  with  equation  (31). 

51.  EXAMPLES.  —  We  will  now  illustrate,  by  numerical  examples, 
the  formulae  for  the  calculation  of  the  variations  of  the  geocentric 
right  ascension  and  declination  arising  from  small  increments  assigned 
to  the  elements.  Let  it  be  required  to  find  for  the  date  1865  Feb- 
ruary 24.5  mean  time  at  Washington,  the  differential  coefficients  of 
the  right  ascension  and  declination  of  the  planet  Eurynome  ©  with 
respect  to  the  elements  of  its  orbit,  using  the  data  and  results  given 
in  Art.  41.  Thus  we  have 

a  =  181°  8'  29".29,      d  =  —  4°  42'  21".56,     log  J  =  0.2450054, 
log  r  =  0.428285,  v  =  129°  3'  50".5,  u  =  326°  41'  40".l, 

A  =  296°  39'  5".0,      B  =  205°  55'  27".l,  C=  212°  32'  17".7, 

log  sin  a  =  9.999716,         log  sin  b  =  9.974825,         log  sin  c  =  9.522219, 
log  x  =  0.425066n,  log  y  =  9.511920,  log  z  =  8.077315, 

e  =  23°  27'  24'7.0,  t—T=  420.714018. 

First,  by  means  of  the  equations  (4),  we  compute  the  following 
values:  — 

log  cos  d  ~  =  8.054308,  log  -^  =  8.668959W, 

log  Cos  d  ^  =  9.754919  ,  log  -^  =  6.968348  , 

dy  dy 

log  ^  —  9.753529. 

Then  we  find  the  differential  coefficients  of  the  heliocentric  co-ordi- 
nates, with  respect  to  TT,  &,  i,  v,  and  r,  from  the  formulae  (7),  which 
give 

ios  =  io    =  °-399496- 


log  --  =  7.876553,     log  --  =  8.830941,    log  --  =  9.222898., 


log  -T-  =  8.726364,      log  --  =  9.687577,    log  --  =  0.142443 


,,, 


log  ~  =--  9.996780n,    log  -^-  =  9.083635,    log  -^  =  7.649030. 


136  THEORETICAL   ASTRONOMY. 

,    dx     dy         ..   dz      . 
In  computing  the  values  of  -TT-,  -^p  and  -TT,  those  of  cos  a,  cos  6, 

and'  cos  c  may  generally  be  obtained  with  sufficient  accuracy  from 
Bin  a,  sin  6,  and  sine.  Their  algebraic  signs,  however,  must  be 
strictly  attended  to.  The  quantities  sin  a,  sin  6,  and  sin  c  are  always 
positive ;  and  the  algebraic  signs  of  cos  a,  cos  6,  and  cos  c  are  indicated 
at  once  by  the  equations  (101)i,  from  which,  also,  their  numerical 
values  may  be  derived.  In  the  case  of  the  example  proposed,  it  will 
be  observed  that  cos  a  and  cos  b  are  negative,  and  that  cos  c  is  positive. 

To  find  the  values  of  cos  d  -j-  and  -7-,  we  have,  according  to  equa- 

(ITC  CfrTT 

tion  (2), 

.  da,  ^  da     dx    ,          .dady  , . .,  N 

cos<5  — =  cos<5- — -fcosd  — -£-,  (41) 

dn  dx     dn  dy     d* 


which  give 


dd  _  dd     dx  dd     dy        dd     dz 

dn  ~  dx     dn  dy     dn        dz     dn' 


=  +  1.42345,  ==_  0.48900. 

n  dv  dn        dv 

In  the  case  of  &,  i,  and  r,  we  write  these  quantities  successively  in 
place  of  Tt  in  the  equations  (41),  and  hence  we  derive 

cos  8  4^  =  -  0-03845,  -^-  =  -  0.09533, 

dQ>  "66 

cos  d  -^-  =  —  0.27641,  -2L  =  —  0.78993, 

ai 


cos  d  ~  =  —  0.08020,  -^-  =  +  0.04873 

Next,  from  (16),  we  compute  the  following  values:  — 

log  ^  =  0.179155,  log  ^-  =  9.577453,  log  ~  =  2.376581,, 

G  dy>  dM0  dp 

log  ^  =  0.171999,  log  -^r  =  9.911247,  log  ~  =  2.535234. 

&  d<f>  dMQ  dfji 

j$y  (j  y 

\Ve  may  now  find  ^—  ,  -^r^,  &c.  by  means  of  the  equations  (11), 

and  thence  the  values  of  cos  d  -=—  ,  -3—,  &c.  :  but  it  is  most  convenient 

d<p  ,  d<p    ' 

to  derive  these  values  directly  from  cos  d  —  =—  ,  cos  d  —  ,—  >  -7-,  and  -7-  » 

dr  dv    dr  dv 

in  connection  with  the  numerical  values  last  found,  according  to  the 


NUMERICAL,   EXAMPLES.  137 

equations  which  result  from  the  analytical  substitution  of  the  expres- 
sions for  -j-,  -j—  >  -y-,  &c.,  in  equation  (2),  writing  successively  <p,  MQ> 
and  fJL  in  place  of  TT.  Thus,  we  have 

.  da,  .da,dr.          .da,     dv 

cos  <5—  -  =  cos  d  —  —  .  —  --  k  cos  <5  -r-  •  -r—  , 
a?'  ar*     a^  av     a?> 

d<5  __  d£     dr        dd     dv 
d<p       dr     dtp       dv     d<? 

and  similarly  for  MQ  and  //,  which  give 

cos  d  -^L  =  +  1.99400,  4^-  =  —  0.65307, 


cos  d    rr  =  +  L13004,  -r  =  —  0.38023, 


cos  5  -^-  =  +  507.264,  ~  =  —  179.315. 

<fc  d/i 

Therefore,  according  to  (1),  we  shall  have 

cos  d  Aa  =  +  1.42345  ATT  —  0.03845  A  ^  —  0.27641  A*    -f  1.99400A^> 

+  1.13004A3f0  -}-  507.264A/., 

A5  =  —  0.48900  ATT  —  0.09533  A  &  —  0.78993Ai    —  0.65307  A£ 

—  0.38023A^f0—  179.315A/Z. 

To  prove  the  calculation  of  the  coefficients  in  these  equations,  we 
assign  to  the  elements  the  increments 

Ajf0  =  +  10",  ATT  ==  -  20",  A  ft  =  -  10",  Ai  =  +  10", 

A^  =  +  10",  A/£  =  +  0".01, 

BO  that  they  become 

Epoch  =  1864  Jan.  1.0  Greenwich  mean  time. 
M0  =     1°  29'  50".21 


TT  =   44    20  13  .09  ^ 

ft  =  206    42  30  .13  >  Mean  Equinox  1864.0 
t  =     4    37     0  .51 J 
V  =   11    16     1  .02 
log  a  =  0.3881288 
L  =  928".56745 


With  these  elements  we  compute  the  geocentric  place  for  1865  Feb- 
ruary 24.5  mean  time  at  Washington ;  and  the  result  is 

a  =  181°  8'  34".81,        d  =  -  4°  42'  30".58,        log  A  =  0.2450284, 


138  THEORETICAL   ASTRONOMY. 

which  are  referred  to  the  mean  equinox  and  equator  of  1865.0.  The 
difference  between  these  values  of  a  and  d  and  those  already  given,  as 
derived  from  the  unchanged  elements,  gives 

Aa  =  +  5".52,  COS  *  Aa  =  +  5".50,  A<S  =  —  9".02, 

and  the  direct  substitution  of  the  assumed  values  of  ATT,  A&,  A*,  &c. 
in  the  equations  for  cos  d  AOC  and  A£,  gives 

cos  d  Aa  =  +  5".46,  Ad  =  —  9".29. 

The  agreement  of  these  results  is  sufficiently  close  to  show  that  the 
computation  of  the  differential  coefficients  has  been  correctly  per- 
formed, the  difference  being  due  chiefly  to  terms  of  the  second  order. 

When  the  differential  coefficients  are  required  for  several  dates,  if 
we  compute  their  values  for  successive  dates  at  equal  intervals,  tho 
use  of  differences  will  serve  to  check  the  accuracy  of  the  calculation  ; 
but,  to  provide  against  the  possibility  of  a  systematic  error,  it  may  be 
advisable  to  calculate  at  least  one  place  directly  from  the  changed 
elements.  Throughout  the  calculation  of  the  various  differential 
coefficients,  great  care  must  be  taken  in  regard  to  the  algebraic  signs 
involved  in  the  successive  numerical  substitutions.  In  the  example 
given,  we  have  employed  logarithms  of  six  decimal  places;  but  it 
would  have  been  sufficient  if  logarithms  of  five  decimals  had  been 
used;  and  such  is  generally  the  case. 

It  will  be  observed  that  the  calculation  of  the  coefficients  of  ATT, 
A  £2,  and  A^  is  independent  of  the  form  of  the  orbit,  depending 
simply  on  the  position  of  the  plane  of  the  orbit  and  on  the  position 
of  the  orbit  in  this  plane.  Hence,  in  the  case  of  parabolic  and 
hyperbolic  orbits,  the  only  deviation  from  the  process  already  illus- 
trated is  in  the  computation  of  the  coefficients  of  the  variations  of 
the  elements  which  determine  the  magnitude  and  form  of  the  orbit 
and  the  position  of  the  body  in  its  orbit  at  a  given  epoch.  In  all 

«.  da          ,.  da    dd        1    dd  . 

cases,  the  values  of  cos  8  -7-,  cos  d-^-t  -3-,  and  -7-  are  determined  as 

dv  dr    dv  dr 

already  exemplified.     If  we  introduce  the  elements  T,  q,  and  e,  we 

shall  have 

do,  do,     dr    .          ,.da,     dv 


dd  _  dd     dr        dd     dv^  .  .~\ 

dT—~dr~"dT  +  ~fo"dt' 

and  similarly  for  the  differential  coefficients  with  respect  to  q  and  e. 


NUMERICAL   EXAMPLES.  139 

„,,  ,       «      ,     ,   . .        ,,          ,  „   dr     dv    dr    dv   dr        .,  dv 

The  mode  of  calculating  the  values  of  -7^,  -777.,  -,-,  -y-,  -j-,  and  -,- 

dT   dT   dq    dq    de  de 

depends  on  the  nature  of  the  orbit. 

In  the  case  of  passing  from  one  system  of  parabolic  elements  to 
another  system  of  parabolic  elements,  the  coefficients  of  Ae  vanish. 

To  illustrate  the  calculation  of  -7™,  -7™,  &c.  in  the  case  of  parabolic 

motion,  let  us  resume  the  values  t — T=  75.364  days,  and  log  q 
=  9.9650486,  from  which  we  have  found 

log  r  =  0.1961120,  v  =  79°  55'  57".26. 

Then,  by  means  of  the  equations  (22),  we  find 

log  |^  =  8.095802n,  log  ~  =  9.242547, 

log  |^=  7.976397tt,  log  ^  =  0.064602n. 

If,  instead  of  dq,  we  introduce  d  log  g,  we  shall  have 

log  -^—  =  9.569812,  log  -TfV~  =  0.391867  . 

fo  d  log  q  &  d  log  q 

From  these,  by  means  of  (43),  we  obtain  the  differential  coefficients 
of  CL  and  d  with  respect  to  T  and  q  or  log  q.  The  same  values  are 
also  used  when  the  variation  of  the  parabolic  eccentricity  is  taken 

into  account.     But  in  this  case  we  compute  also  -7-  from  equation 
dr 
de 


(31)  and  f-  from  (33)  or  (34),  which  give,  for  v  =  79°  55'  57".3, 


log  ~  =  8.147367  ,  log  ^  =  9.726869. 

de  de 

In  the  case  of  very  eccentric  orbits,  the  values  of  -77^,  -r^  &c.  are 
found  from 


dv  klp  dr  k       .  ,     . 

(44) 


. 

dq  qr*  dq        q 

dr  _  r   .  r*esmv    dv 

dq  q  p         dq 

the  mass  being  neglected. 


140  THEOEETICAL   ASTKONOMY. 

To  illustrate  the  application  of  these  formulae,  let  us  resume  the 
values,  t—T=  68.25  days,  e  =  0.9675212,  and  log q  =  9.7668134, 
from  which  we  have  found  (Art.  41) 

v  =  102°  20'  52".20,  log  r  =  0.1614052. 

Hence  we  derive 

logjp  =  0.0607328, 

and 

5 


log— =  7.943137n,  log^,=  8.180711., 


log  -     =  0.186517.,  log       =  0.186517  . 

dq  dq 

If  we  wish  to  obtain  the  differential  coefficients  of  v  and  r  with 
respect  to  log  q  instead  of  q,  we  have 

dv      _q    dv  dr      _q    dr 

dlogq       Afl    dq  dlogq      A0    dq' 

in  which  ^0  is  the  modulus  of  the  system  of  logarithms. 

Then  we  compute  the  value  of  -7-  by  means  of  the  equation  (30). 
(35),  (39),  or  (40).  The  correct  value  as  derived  from  (39)  is 

^  =  -0.24289. 
de 

The  values  derived  from  (35),  omitting  the  last  term,  from  (40)  and 
from  (30),  are,  respectively,  —  0.24440,  —  0.24291,  and  —  0.23531. 
The  close  agreement  of  the  value  derived  from  (40)  with  the  correct 
value  is  accidental,  and  arises  from  the  particular  value  of  v,  which 
is  here  such  as  to  make  the  assumptions,  according  to  which  equation 
(40)  is  derived  from  (39),  almost  exact. 

Finally,  the  value  of  -7-  may  be  found  by  means  of  (32),  whicn 
gives 

^  =  +  0.70855. 
de 

When,  in  addition  to  the  differential  coefficients  which  depend  on 
the  elements  T,  q,  and  e,  those  which  depend  on  the  position  of  the 
orbit  in  space  have  been  found,  the  expressions  for  the  variation  of 
the  geocentric  right  ascension  and  declination  become 


NUMERICAL   EXAMPLES.  141 

COS  <5  Aa  =  COS  £  -^  ATT  -f-  COS  d  —  —  A  &  +  COS  d  ~  Az  -j-  COS  d  —  J*  A  I7 

«7r  aQ  di  dT 

.da  da, 

-f  cos  d  —  A0  -f  cos  <5  -i-  Ae, 
dq  de 

dd  dd  dd  dd 


If  we  introduce  log*/  instead  of  q,  the  terms  containing  q  become 
respectively  cos  d  A  log  7  and  -yj  —  A  log  7.  It  should  be 

observed  that  if  ATI,  A&,  and  A?'  are  expressed  in  seconds,  in  order 
that  these  equations  may  be  homogeneous,  the  terms  containing  AT, 
A<?,  and  Ae  must  be  multiplied  by  206264.8;  but  if  ATT,  A&,  and  AZ 
are  expressed  in  parts  of  the  radius  as  unity,  the  resulting  values  of 
cos  d  Aa  and  A£  must  be  multiplied  by  206264.8  in  order  to  express 
them  in  seconds  of  arc. 

The  most  general  application  of  the  equations  for  cos  d  Aa  and  A£ 
in  terms  of  the  variations  of  the  elements  is  for  the  cases  in  which 
the  values  of  cos  d  Aa  and  of  A<5  are  already  known  by  comparison 
of  the  computed  place  of  the  body  with  the  observed  place,  and  in 
which  it  is  required  to  find  the  values  of  ATT,  A&,  Ai,  &c.,  which, 
being  applied  to  the  elements,  will  make  the  computed  and  the 
observed  places  agree.  When  the  variations  of  all  the  elements  of 
the  orbit  are  taken  into  account,  at  least  six  equations  thus  derived 
are  necessary,  and,  if  more  than  six  equations  are  employed,  they 
must  first  be  reduced  to  six  final  equations,  from  which,  by  elimina- 
tion, the  values  of  the  unknown  quantities  ATT,  A&,  &c.  may  be 
found.  In  all  such  cases,  the  values  of  Aa  and  A£,  as  derived  from 
the  comparison  of  the  computed  with  the  observed  place,  are  ex- 
pressed in  seconds  of  arc;  and  if  the  elements  involved  are  expressed 
in  seconds  of  arc,  the  coefficients  of  the  several  terms  of  the  equations 
must  be  abstract  numbers.  But  if  some  of  the  elements  are  not 
expressed  in  seconds,  as  in  the  case  of  T,  q,  and  e,  the  equations 
formed  must  be  rendered  homogeneous.  For  this  purpose  we  mul- 
tiply the  coefficients  of  the  variations  of  those  elements  which  are 
not  expressed  in  seconds  of  arc  by  206264.8.  Further,  it  is  gene- 
rally inconvenient  to  express  the  variations  AT,  Ag,  and  Ae  in  parts 
of  the  units  of  T,  q,  and  e,  respectively  ;  and,  to  avoid  this  incon- 
venience, we  may  express  these  variations  in  terms  of  certain  parts 
of  the  actual  units.  Thus,  in  the  case  of  T,  we  may  adopt  as  the 
mat  of  AT  the  nth  part  of  a  mean  solar  day,  and  the  coefficients 
of  the  terms  of  the  equations  for  cos<5  Aa  and  A£  which  involve  A? 


142  THEORETICAL   ASTRONOMY. 

must  evidently  be  divided  by  n.  In  the  same  manner,  it  appears 
that  if  we  adopt  as  the  unit  of  A<?  the  unit  of  the  rath  decimal 
place  of  its  value  expressed  in  parts  of  the  unit  of  q,  we  must  divide 
its  coefficient  by  10™,  and  similarly  in  the  case  of  Ae,  so  that  the 
equations  become 

COS  d  Aa  =  COS  S  —  ATT  -j-  COS  3  -  —  A  &  -j-  COS  8  —  -  Al  -f  -  COS  d  —  -  A  T 

d*  dQ  di  n          dT 

S  .  da  S  .da,  ,  .  _,. 

Ae' 


dd  dd  dd  s     dS  s    d3 

=  —  ATT  +  —  A  £  +  TV  ^  +  -  .  —  A  T  +  —  — 

dn  d&  di  n    dT  10m  dq 

s    dd 


in  which  s  =  206264.8.  When  log  q  is  introduced  in  place  of  q,  the 
coefficients  of  A  log  q  are  multiplied  by  the  same  factor  as  in  the  case 
of  4*7,  the  unit  of  A  log  q  being  the  unit  of  the  rath  decimal  place 
of  the  logarithms.  The  equations  are  thus  rendered  homogeneous, 
and  also  convenient  for  the  numerical  solution  in  finding  the  values 
of  the  unknown  quantities  ATT,  A&,  A^,  AT,  &c.  When  AT,  Ag,  and 
Ae  have  been  found  by  means  of  the  equations  thus  formed,  the 

A  HP       A  /> 

coirections  to  be  applied  to  the  corresponding  elements  are  —  ,  y^, 

and  T7y^7-     In  the  same  manner,  we  may  adopt  as  the  unknown 

quantity,  instead  of  the  actual  variation  of  any  one  of  the  elements 
of  the  orbit,  n  times  that  variation,  in  which  case  its  coefficient  in 
the  equations  must  be  divided  by  n. 

The  value  of  Aa,  derived  by  taking  the  difference  between  the 
computed  and  the  observed  place,  is  affected  by  the  uncertainty 
necessarily  incident  to  the  determination  of  a  by  observation.  The 
unavoidable  error  of  observation  being  supposed  the  same  in  the  case 
of  a  as  in  the  case  of  d,  when  expressed  in  parts  of  the  same  unit, 
it  is  evident  that  an  error  of  a  given  magnitude  will  produce  a 
greater  apparent  error  in  a  than  in  o,  since  in  the  case  of  a  it  is 
measured  on  a  small  circle,  of  which  the  radius  is  cos  d  •  and  hence, 
in  order  that  the  difference  between  computation  and  observation  in 
a  and  d  may  have  the  same  influence  in  the  determination  of  the 
corrections  to  be  applied  to  the  elements,  we  introduce  COS^AOC 
instead  of  Aa.  The  same  principle,  is  applied  in  the  case  of  the 
longitude  and  of  all  corresponding  spherical  co-ordinates. 


DIFFERENTIAL   FORMULAE.  143 

52.  The  formulae  already  given  will  determine  also  the  variations 
of  the  geocentric  longitude  and  latitude  corresponding  to  small  in- 
crements assigned  to  the  elements  of  the  orbit  of  a  heavenly  body. 
In  this  case  we  put  e  =  0,  and  compute  the  values  of  A,  I>,  sin  a, 
and  sin  b  by  means  of  the  equations  (94)r  We  have  also  (7=0, 
sin  c  =  sin  i,  and,  in  place  of  a  and  d,  respectively,  we  write  ^  and  /?. 
But  when  the  elements  are  referred  to  the  same  fundamental  plane 
as  the  geocentric  places  of  the  body,  the  formulae  which  depend  on 
the  position  of  the  plane  of  the  orbit  may  be  put  in  a  form  which  is 
more  convenient  for  numerical  application. 

If  we  differentiate  the  equations 

xf  =  r  cos  u  cos  ft  —  r  sin  u  sin  ft  cos  it 

y'  =  r  cos  u  sin  ft  -f-  r  sin  u  cos  ft  cos  *, 
z'  =  r  sin  u  sin  i, 
we  obtain 

dxr  =  —dr  —  r  (sin  u  cos  ft  -f-  cos  u  sin  ft  cos  i)  du 
—  r  (cos  u  sin  ft  -f  sin  u  cos  ft  cos  i)  dft  -f-  r  sin  u  sin  ft  sin  i  cfo', 

dtf  =  —dr  —  r  (sin  u  sin  &  —  cos  u  cos  &  cos  i)  du 
-f  r  (cos  tt  cos  ft  —  sin  w  sin  ft  cos  i)  c?ft  —  r  sin  w  cos  ft  sin  i  eft,  (46) 

dz1  =  -  dr  -{•  r  cos  u  sin  idu  -\-  r  sin  it  cos  i  eft, 
r 

in  which  #',  £/',  z'  are  the  heliocentric  co-ordinates  of  the  body  in 
reference  to  the  ecliptic,  the  positive  axis  of  x  being  directed  to  the 
vernal  equinox.  Let  us  now  suppose  the  place  of  the  body  to  be 
referred  to  a  system  of  co-ordinates  in  which  the  ecliptic  remains  as 
the  plane  of  xy,  but  in  which  the  positive  axis  of  x  is  directed  to  the 
point  whose  longitude  is  ft ;  then  we  shall  have 

dx  =  dx'  cos  &  +  dtf  sin  & , 
dy  =  —  dx'  sin  &  +  dtf  cos  ft , 
dz  =  dz', 

and  the  preceding  equations  give 

dx  =  -  dr  —  r  sin  u  du  —  r  sin  u  cos  i  d& , 

T 

dy  =  -  dr  +  r  cos  u  cos  i  du  -f-  r  cos  u  d&  —  r  sin  it  sin  i  di,    (47) 
dz  =  _  dr  -f  r  cos  u  sin  idu  ~{-r  sin  ?f  cos  i  eft. 


144  THEORETICAL   ASTRONOMY. 

This  transformation,  it  will  be  observed,  is  equivalent  to  diminishing 
the  longitudes  in  the  equations  (46)  by  the  angle  &  through  which 
the  axis  of  x  has  been  moved. 

Let  Xn  Y,,  Z,  denote  the  heliocentric   co-ordinates  of  the  earth 
referred  to  the  same  system  of  co-ordinates,  and  we  have 

x  -f-  X,  =  A  cos/3  cos  0*  —  Q), 
V  +  F,=  Jcos08in(J  —  £), 
z  -f  Z,  =  A  sin  /?, 

in  which  ^  is  the  geocentric  longitude  and  /9  the  geocentric  latitude. 
In  differentiating  these  equations  so  as  to  find  the  relation  between 
the  variations  of  the  heliocentric  co-ordinates  and  the  geocentric  lon- 
gitude and  latitude,  we  must  regard  Q,  as  constant,  since  it  indicates 
here  the  position  of  the  axis  of  x  in  reference  to  the  vernal  equinox, 
and  this  position  is  supposed  to  be  fixed.  Therefore,  we  shall  have 


—     cos    sn 

dy=  cos  £  sin  (A  —  &)d/f  —  A  sin  /5  sin  (A  —  £)d£-|-  A  cos/?  cos  (A 
cfe  =sin  /?  dJ  H-  J  cos  y9  d/9, 


from  which,  by  elimination,  we  find 


sin/?  cos  (A—  a)  ,       sin  /9  sin  (A—  q)  cos  /? 


--  _ 
These  equations  give 


COS  p  —  =  —  == 

r 


—  =  —          --  -.  -  .  —j— 

dx  A  dx 


Q  ft 

cosj?  -j-  =  0,  -,- 

C?3  ^2 

If  we  introduce  the  distance  o  between  the  ascending  node  and  the 
place  of  the  perihelion  as  one  of  the  elements  of  the  orbit,  we  have 

du  =  dv  -{-  d<0, 
and  the  equations  (47)  give 

dx       x  dy       y  dz       z         .         .    . 

—~-=-  =  cosu,          —/-  =  -==  smu  cos  i,      —=—  =-  =  smw  smt; 
dr        r  dr       r  dr       r 

dx        dx  dy        dy  .     dz        dz 

—=—  =  -_—  =  —  rsmu,  —.y—  =  -.y-  =r  cos  u  cost,  —=—  =  -=— 

av       aw  dv       dw  dv       dot 


DIFFERENTIAL   FORMULA.  145 

^-=rcos«,  -*l=0j  (49) 

dx       A  dy  .    .         c?z 

7.   =  0.  — =r-  =  —  ?*  sin  w  sin ^.      — ^r-  =  r  sin u  cos i. 

at  at  di 

If  we  introduce  TT,  the  longitude  of  the  perihelion,  we  have 
du  =  dv  -{-  dit  — •  a*  & , 

and  hence  the  expressions  for  the  partial  differential  coefficients  of 
the  heliocentric  co-ordinates  with  respect  to  it  and  &  become 


dx  dy  dz 

— r= —  —  —  r  sm  w,  7     —  r  cos  u  cos  ^,         — , —  =  r  cos  w  sin  i ; 

f  f  t  (») 

dx         _  .  , , .        aw       •  .  .  t .        az 

-— —  =  2r  sm  w  sm2  Jt,     ~-  =  2r  cos  w  sm2  J*,     -7—  =  —  r  cos  w  sin  i. 

When  the  direct  inclination  exceeds  90°  and  the  motion  is  regarded 
as  being  retrograde,  we  find,  by  making  the  necessary  distinctions  in 
regard  to  the  algebraic  signs  in  the  general  equations, 

dx  dy  dz 

-rr  =  0,         -T-  =  f  sin  u  sin  i,        — j-r  =  —  r  sin  u  cos  i ;      (51) 

di  di  di 

„      dx    dx      dx     dy    0  , 

and  the  expressions  for  -7-,  -j-,  -TQ->  -f-,  &c.  are  derived  directly 

from  (49)  by  writing  180°  —  i  in  place  of  i.     If  we  introduce  the 
longitude  of  the  perihelion,  we  have,  in  this  case, 

du  =  dv  —  oV  +  d  Q , 
and  hence 

dx  dy  dz  .    . 

— /-  =  r  cos  i*  cos  i,       -j —  =  —  rcosttsmi; 


(52) 
-r~-  =  —  2r  sin  w  sin2  £i,  -~-  =  2r  cos  tt  sin2  Ji,  -^— -  =  r  cos  w  sin  i. 

But,  to  prevent  confusion  and  the  necessity  of  using  so  many  for- 
mulae, it  is  best  to  regard  i  as  admitting  any  value  from  0°  to  180°, 
and  to  transform  the  elements  which  are  given  with  the  distinction 
of  retrograde  motion  into  those  of  the  general  case  by  taking 
180°  —  i  instead  of  i,  and  2&  — n  instead  of  TT,  the  other  elements 
remaining  the  same  in  both  cases. 

53.  The  equations  already  derived  enable  us  to  form  those  for  the 
differential  coefficients  of  ^  and  ft  with  respect  to  r,  v,  & ,  i,  and  co  or 
T,  by  writing  successively  X  and  ft  in  place  of  0,  and  &,  i,  &c.  in 


10 


146  THEORETICAL,   ASTRONOMY. 

place  of  it  in  equation  (2).  The  expressions  for  the  differential  coeffi- 
cients of  r  and  v,  with  respect  to  the  elements  which  determine  the 
form  of  the  orbit  and  the  position  of  the  body  in  its  orbit,  being 
independent  of  the  position  of  the  plane  of  the  orbit,  are  the  same  as 
those  already  given;  and  hence,  according  to  (42)  and  (43),  we  may 
derive  the  values  of  the  partial  differential  coefficients  of  A  and  /9 
with  respect  to  these  elements.  The  numerical  application,  however, 
is  facilitated  by  the  introduction  of  certain  auxiliary  quantities. 
Thus,  if  we  substitute  the  values  given  by  (48)  and  (49)  in  the 
equations 

.  c?A  ctt     d x    .         Qctt     dy 

cos  /9  -j-  =  cos  /?  -=-  •  -j — f-  cos  /5—-  •  -/-, 
dv  dx     dv    l  dy     dv 

&L—&L  ^L\~  ty-+dP-  — 

dv        dx     dv       dy     dv        dz     dv' 
and  put 

cos  i  cos  (A  —  &  )  =  AQ  sin  A, 
sin  (A  —  &  )  =  AQ  cos  A, 

sin  i  =  n  sin  N, 
—  sin  (A  —  £2  )  cos  i  =  n  cos  N, 

in  which  AQ  and  n  are  always  positive,  they  become 


dv  dw       A 

-~  =          —-==  —  (sin  /9  cos  (A  —  &  )  sin  u  -f-  n  cos  u  sin  (N  +  ft  ). 

Let  us  also  put 

n  sin  ( N  +  ft  =  -B0  gin  -^> 
sin  /5  cos  (A  —  &  )  =  J?0  cos  jB, 
and  we  have 

cos  /?  -j—  =  cos  /?  -j—  =  —  ^40  sin  (A  -\-  u  -, 

*          ,!  CM) 


7  •»  TO 

The  expressions  for  cos  ft  -j-  and  -j-  give,  by  means  of  the  same 
auxiliary  quantities, 

cos  0  -= —  = ^  cos  (J.  +  u), 

dr  (56) 

f  =  -^eos(*  +  «). 

In  the  same  manner,  if  we  put 


DIFFERENTIAL   FORMULA.  147 

cos  (/I  —  £)=  C0sin  C, 

cos  i  sin  (A  —  &  )  =  (7,  cos  (7; 

1  67) 
cos  i  =  D  sin  Z> 


sin  f  A  —  &)  sin  i  =  D0  cos  D; 
we  obtain 

-^-  =  =  ^  (70  sin 


A  r 

cos  /9  —  JT-  =  —  -j  sin  i  sin  w  cos  (A  —  &  ), 

-2(L  =  -Dfl  sin  w  sin  (D  +  £). 

Cd  T 

If  we  substitute  the  expressions  (55)  and  (56)  in  the  equations 


fl  .  .         fl 

cos  /9  3—  =  cos  /?  -=-  •  3  --  ^-  cos  /?  -j-  •  3— 
d?  dr    dy  dv    d<f> 


and  put 


__  _       ^._ 

d<p  ~  dr    dy        dv 


dr       /.  .    rr 
—  j—  =/  sin  F  =  a  cos  ^  cos  v, 


r  j—  =/  cos  F=  I  --  f-  tan  <p  cos  v  I  r  sin  v, 
d^      ^  \cos?   '  / 


.*L  =  feMp=lJ_ 
we  get 

JF+«), 


In  a  similar  manner,  if  we  put 

dr  „ 

--  TVF  =  a  sin  W  =  —  a  tan  ^>  sin  v, 


dv  ~       a2  cos  <f> 

-- 


T 

—  --  =  h  sinH=  — 


9r  \ 

smv(t—  T)  —  y-  206264.8  J, 


c£y        ,         ,-..      a*  cos  <P  ,.       ...x 
r  -T—  =  &  cos  IT=  -  —  (t  —  T\ 
dp.  r 


148  THEORETICAL  ASTRONOMY. 

•we  obtain 


di        h  (62) 

cos/9  JjL  =  *  J.  sin  (^  +  Jff  +  «), 


The  quadrants  in  which  the  auxiliary  angles  must  be  taken  are 
determined  by  the  condition  that  A0)  BQ,  C0,  /,  g,  and  h  are  always 
positive. 

54.  If  the  elements  T,  q,  and  e  are  introduced  in  place  of  Jf0,  //, 
and  ^>,  we  must  put 


=r-,,  (63) 


--  -j-,  -5- 

aq  aq 

and  the  equations  become 


=          sn 


«S       9  (64) 

ooi^-^iiaU  +  Jr+ii), 

4^  =  \  B0  sin  (jB  H-  H+  u). 

In  the  numerical  application  of  these  formulae,  the  values  of  the 
second  members  of  the  equations  (63)  are  found  as  already  exem- 
plified for  the  cases  of  parabolic  orbits  and  of  elliptic  and  hyperbolic 
orbits  in  which  the  eccentricity  differs  but  little  from  unity.  In  the 
same  manner,  the  differential  coefficients  of  A  and  ft  with  respect  to 
any  other  elements  which  determine  the  form  of  the  orbit  may  be 
computed. 


NUMERICAL   EXAMPLES.  149 

In  the  case  of  a  parabolic  orbit,  if  the  parabolic  eccentricity  is 
supposed  to  be  invariable,  the  terms  involving  e  vanish.  Further, 
in  the  case  of  parabolic  elements,  we  have 

.     ~  dr       ksmv  dv 

=   -- 


which  give 

tan  G  =  —  tan 


V2 
-,  which  is  the 

expression  for  the  linear  velocity  of  a  comet  moving  in  a  parabola. 
Therefore, 


<W  kl/2 

AQ  sin  (A  -f-  u  —  %v), 

(65) 


cos  ^  dT  =  ~~  A  °  sin  (A  +  u  ~ 


For  the  case  in  which  the  motion  is  considered  as  being  retrograde, 
180°  — i  must  be  used  instead  of  i  in  computing  the  values  of  Aw 
A,  n,  N,  C0,  and  (7,  and  the  equations  (55),  (56),  and  the  first  two 
of  (58),  remain  unchanged.  But,  for  the  differential  coefficients  with 
respect  to  i,  the  values  of  DQ  and  D  must  be  found  from  the  last  two 
of  equations  (57),  using  the  given  value  of  i  directly ;  and  then  we 
shall  have 

dk       r 
cos  /?  -TV-  =  -7  sin  i  sin  u  cos  (A  —  &  ), 

M 

w  =  —  L  DQ  gin  u  sm  (j)  _|_  ft. 

55.  EXAMPLES. — The  equations  thus  derived  for  the  differential 
coefficients  of  X  and  /9  with  respect  to  the  elements  of  the  orbit, 
referred  to  the  ecliptic  as  the  fundamental  plane,  are  applicable  when 
any  other  plane  is  taken  as  the  fundamental  plane,  if  we  consider  ^ 
and  /9  as  having  the  same  signification  in  reference  to  the  new  plane 
that  they  have  in  reference  to  the  ecliptic,  the  longitudes,  however, 
being  measured  from  the  place  of  the  descending  node  of  this  plane 
on  the  ecliptic.  To  illustrate  their  numerical  application,  let  it  be 
required  to  find  the  differential  coefficients  of  the  geocentric  right 
ascension  and  declination  of  Eurynome  ©  with  respect  to  the  ele- 
ments of  its  orbit  referred  to  the  equator,  for  the  date  1865  February 
24.5  mean  time  at  Washington,  using  the  data  given  in  Art.  41. 


150  THEORETICAL  ASTRONOMY  , 

In  the  first  place,  the  elements  which  are  referred  to  the  ecliptic 
must  be  referred  to  the  equator  as  the  fundamental  plane  ;  and,  by 
means  of  the  equations  (109)x,  we  obtain 


£'=353°  45'  35".87,        i'  =  19°  26'  25".76,        o>0  =  212°  32'  17".71, 

and 

a/  ==  w  +  a>0  =  50°  10'  7".29, 

which  are  the  elements  which  determine  the  position  of  the  orbit  in 
space  when  the  equator  is  taken  as  the  fundamental  plane.  These 
elements  are  referred  to  the  mean  equinox  and  equator  of  1865.0. 
Writing  a  and  d  in  place  of  ^  and  /9,  and  &',  if,  wf  in  place  of  &,  i, 
and  co,  respectively,  we  have 

AQ  sin  A  —  cos  (<*—&')  cos  ir,  A0  cos  A  =  sin  (a  —  &')  ; 

n  sin  N=  sin  if,  n  cosN=  —  cost'  sin  (a  —  &'); 

J50  sin  B  =  n  sin  (N  +  #)»  BQ  cos  -B  =  sin  <5  cos  (a  —  &')  ; 

<70  sin  0=  cos  (a  —  &'),  ^o  c<>s  C^sin  (a  —  &')  cosi'; 

A  sin  Z>  =  cos  i',  D0  cos  D  =  sin  i'  sin  (a  —  &  ')  ; 
/  sin  F=a  cos  ^  cos  v, 

/    2  \ 

/  cos  F=  1  --  h  tan  ?  cos  v  }  r  sin  v; 
Vcos?'  / 

gsmG  =  —  a  tan  <?  sin  v, 


—  T)  —  --  206264.8   , 


The  values  of  Aw  n,  BQ,  Cw  D0,  /,  g9  and  h  must  always  be  positive, 
thus  determining  the  quadrants  in  which  the  angles  A,  B,  &c.  must 
he  taken  ;  and  these  equations  give 

log  A0  =  9.97497,  A  =  262°  10'  40", 

log  B0=  9.52100,  B=   75    48  35  , 

log  C0  =  9.99961,  (7  =  263     26, 

log  D0  =  9.97497,  D=   92    3547, 

log/  =0.62946,  .F  =  339    14     0, 

log  <y  =0.34593,  £  =  350    11  16, 

log  A  =  2.97759,  ,     H=   14    30  48  , 
uf  =  v      oi'  =  179°  13'  58". 


NUMERICAL   EXAMPLES.  151 

Substituting  these  values  in  the  equations  (55),  (58),  (60),  and  (62), 
and  writing  a  and  d  instead  of  ^  and  /?,  and  u'  in  place  of  u,  we  find 

'  -  +  1.4235,  j,  =  _  0.4890, 

~ 


cos  »  ~T==  +  1.5098,  -,  =  +  0.0176, 


cos  d  -^  =  4  0.0067,  -!rr  =  +  0.0193, 

v**  tti 

cos  5  -^-  ==  -f- 1.9940,  -~-  =  —  0.6530, 

-~r  =  _  0.3802, 

-^j-  =  4  507.25,  ~  =  —  179.34 ; 

and  hence 

cos  d  Aa  =  4- 1.4235  AW'  4  1.5098  A&'  4  0.0067  tf  4  1.9940  A? 

4- 1.1300  Aif0  +  507.25  AAI, 
A<?  =  —  0.4890  AW'  +  0.0176  A&'  4  0.0193  Ai'  —  0.6530  A? 

-  0.3802  A  Jf0  — 179.34  AA*. 
If  we  put 

A«/=r  — 6".64,  *&'=  — 14".12,  AI*=  — 8".86, 

A??  =  4  10",  AlT0  =  4  10",  A/*  =  4  0".01, 

we  get 

cos  5  Aa  =  4  5".47,  A<5  =  —  9".29 ; 

and  the  values  calculated  directly  from  the  elements  corresponding  to 
the  increments  thus  assigned,  are 

cos  d  Aa  =  4-  5".50,  A<5  =  —  9".02. 

The  agreement  of  these  results  is  sufficiently  close  to  prove  the  cal- 
culation of  the  coefficients  in  the  equations  for  cos  d  AOC  and  A£. 

When  the  values  of  AW',  A  & r,  and  Air  are  small,  the  correspond- 
ing values  of  AW,  A&,  and  A^  may  be  determined  by  means  of 
differential  formulae.  From  the  spherical  triangle  formed  by  the 
intersection  of  the  planes  of  the  orbit,  ecliptic,  and  equator  with  the 
celestial  vault,  we  have 

cos  ^  =  cos  i'  cos  e  4  sin  i'  sin  e  cos  & ', 
sin  i  cos  &  =  —  cos  i'  sin  e  4-  sin  i'  cos  e  cos  Q>f, 
sin  i  sin  &  =  sin  i'  sin  &  f> 
sin  i  sin  w0  =  sin  & '  sin  e, 
sin  i  cos  w0  =  cos  £  sin  i'  —  sin  e  cos  i'  cos  & ', 


152  THEORETICAL   ASTRONOMY. 

from  which  the  values  of  &,  i,  and  co0  may  be  found  from  those  of 
& '  and  ir.  If  we  differentiate  the  first  of  these  equations,  regarding 
e  as  constant,  and  reduce  by  means  of  the  other  given  relations,  we 
get 

di  =  cos  to0  di'  -f-  sin  a>0  sin  i'  d  & '.  (68) 

Interchanging  i  and  180°  —  i',  and  also  &  and  &',  we  obtain 

di'  =  cos  tt>0  di  —  sin  w0  sin  idQ. 
Eliminating  di  from  these  equations,  and  introducing  the  value 

sin  i' sin  & 

sini       sin&y' 
the  result  is 

sin  Q  sin  toK 


Biny  ami 

If  we  differentiate  the  expression  for  cos  co0  derived  from  the  same 
spherical  triangle,  and  reduce,  we  find 

da>0  —  cos  i  d£l  —  cos  i'  dQ'. 

Substituting  for  dQ,  its  value  given  by  the  preceding  equation,  and 
reducing  by  means  of 

sin  & '  cos  i'  =  sin  &  cos  w0  cos  i  —  cos  Q,  sin  w0, 
we  get 

sin  tn  ein  m 

(70) 


The  equations  (68),  (69),  and  (70)  give  the  partial  differential  co- 
efficients of  &  ,  i,  and  w0  with  respect  to  &  ;  and  i',  and  if  we  sup- 
pose the  variations  of  the  elements,  expressed  in  parts  of  the  radius 
as  unity,  to  be  so  small  that  their  squares  may  be  neglected,  we  shall 

have 

sin  ojn 

*^=Esf'«" 

sin 


At  =  sin  ta0  sin  i'  A  Qf  -f-  cos  %  Air, 


If  we  apply  these  formulae  to  the  case  of  Eurynome,  the  result  is 


WO  =  —  4.420A&'  + 

&  =  _  3.488A  Q'  +  6.686A/, 

Ai  =  —  0.179A^'  —  0.843  AI*  ; 


DIFFERENTIAL   FORMULAE.  153 

and  if  we  assign  the  values 

A£'  =  —  14".12,  Ai'  =  —  8".86,  AC//  =  —  6".64, 

we  get 
AOIO  =  +  3".36,       A  ^  =  —  10".0,       Ai  =  +  10".0,       A«>  =  —  10".0, 

and,  hence,  the  elements  which  determine  the  position  of  the  orbit  in 
reference  to  the  ecliptic. 

The  elements  a/,  &  ',  and  if  may  also  be  changed  into  those  for 
which  the  ecliptic  is  the  fundamental  plane,  by  means  of  equations 
which  may  be  derived  from  (109)!  by  interchanging  &  and  &'  and 
180°  —  i'andt. 

56.  If  we  refer  the  geocentric  places  of  the  body  to  a  plane  whose 
inclination  to  the  plane  of  the  ecliptic  is  iy  and  the  longitude  of  whose 
ascending  node  on  the  ecliptic  is  &,  —  which  is  equivalent  to  taking 
the  plane  of  the  orbit  corresponding  to  the  unchanged  elements  as 
the  fundamental  plane,  —  the  equations  are  still  further  simplified. 
Let  x'j  y'j  zf  be  the  heliocentric  co-ordinates  of  the  body  referred  to 
a  system  of  co-ordinates  for  which  the  plane  of  the  unchanged  orbit 
is  the  plane  of  xy,  the  positive  axis  of  x  being  directed  to  the  as- 
cending node  of  this  plane  on  the  ecliptic;  and  let  x,  y,  z  be  the 
heliocentric  co-ordinates  referred  to  a  system  in  which  the  plane  of 
xy  is  the  plane  of  the  ecliptic,  the  positive  axis  of  x  being  directed 
*o  the  point  whose  longitude  is  Q  .  Then  we  shall  have 


dtf  =  dy  cos  i  -f-  dz  sin  it 
dz'  =  —  dy  sin  i  -f-  dz  cos  i. 

Substituting  for  dx,  dyy  and  dz  their  values  given  by  the  equations 
(47),  we  get 

xr 
dxf  =  —  dr  —  r  sin  u  du  —  r  sin  u  cos  i  dQ, 

dy'  =  —  dr  -f-  r  cos  u  du  -\-  r  cos  u  cos  i  d&y 
dz'  =  —  dr  —  r  cos  u  sin  i  dQ>  +  r  sul  u  di* 

It  will  be  observed  that  we  have,  so  long  as  the  elements  remain 
unchanged, 

a/  =  r  cos  u,  yf  =  r  sin  u,  z'  =  0, 


154  THEOKETICAL   ASTRONOMY. 

and  hi.'nce,  omitting  the  accents,  so  that  x9  y,  z  will  refer  to  the  plane 
of  the  unchanged  orbit  as  the  plane  of  xy,  the  preceding  equations 
give 

dx  =  cos  u  dr  —  r  sin  u  du  —  r  sin  u  cos  i  dQ  , 
dy  =  sin  u  dr  -\-  r  cos  u  du  -j-  r  cos  u  cos  i  dQ  , 
dz  —  —  r  cos  u  sin  ^  dQ>  -f-  r  sin  i*  cfo*. 

The  value  of  ^>  is  subject  to  two  distinct  changes,  the  one  arising 
fiom  the  variation  of  the  position  of  the  orbit  in  its  own  plane,  and 
the  other,  from  the  variation  of  the  position  of  the  plane  of  the  orbit. 
Let  us  take  a  fixed  line  in  the  plane  of  the  orbit  and  directed  from 
the  centre  of  the  sun  to  a  point  the  angular  distance  of  which,  back 
from  the  place  of  the  ascending  node  on  the  ecliptic,  we  shall  desig- 
nate by  ff-  and  let  the  angle  between  this  fixed  line  and  the  semi- 
transverse  axis  be  designated  by  £.  Then  we  have 

X  =  a>  -f  ff. 

The  fixed  line  thus  taken  is  supposed  to  be  so  situated  that,  so  long 
as  the  position  of  the  plane  of  the  orbit  remains  unchanged,  we  have 


But  if  the  elements  which  fix  the  position  of  the  plane  of  the  orbit 
are  supposed  to  vary,  we  have  the  relations 

dff  =  cosi  d£l, 

da>  =  d%  —  cos  i  d&  ,  (72) 

dn  =  dx  +  (1  —  cosi)  d&  =  dx  +  2  sm*±i  d&. 

Now,  since  u  =  v  -f-  to,  we  have 

u=v+X—ff* 

and 

du  —  dv  -f-  dx  —  dff  =  dv  -f-  d%  —  cos  i  d  &  • 

Substituting  this  value  of  du  in  the  equations  for  dx,  dy,  dz,  they 

reduce  to 

dx  =  cos  u  dr  —  r  sin  u  dv  —  r  sin  u  dx, 

dy  =  sin  u  dr  -f  r  cos  u  dv  +  r  cos  u  d%,         (73) 

dz  =  —  r  cos  u  sin  i  dQ,  -j-  r  sin  u  di. 

The  inclination  is  here  supposed  to  be  susceptible  of  any  value  from 
0°  to  180°,  and  if  the  elements  are  given  with  the  distinction  of 
retrograde  motion  we  must  use  180°  —  i  instead  of  i. 

Let  us  now  denote  by  6  the  geocentric  longitude  of  the  body  mea- 
sured in  the  plane  of  the  unchanged  orbit  (which  is  here  taken  as  the 


DIFFERENTIAL   FORMULA. 


155 


fundamental  plane)  from  the  ascending  node  of  this  plane  on  the 
ecliptic,  and  let  the  geocentric  latitude  in  reference  to  the  same  plane 
be  denoted  by  y.  Then  we  shall  have 

x  -f-  X—  A  cos  y  cos  0, 
y  +  Y=  A  cos  y  sin  0, 
z  +  Z  =  A  sin  y, 

in  which  X,  F,  Z  are  the  geocentric  co-ordinates  of  the  sun  referred 
to  the  same  system  of  co-ordinates  as  x,  y}  and  z.  These  equations 
give,  by  differentiation, 

dx  =  cos  y  cos  6  d  A  —  A  sin  y  cos  0  dy  —  A  cos  y  sin  0  dOt 
dy  =  cos  y  sin  6  d  A  —  A  sin  y  sin  0  dfy  -f~  ^  cos  ^  cos  ^  ^> 
efe  =  sin  y  dA  -\-  A  cos  ^  cfy ; 

and  hence  we  obtain 

7 .           sin  0  ,     .   cos  0 
cos  y  do  = —  dx  -\ — -.—  dy, 

,          _  sin  y  cos  0  ,     __  sin  17  sin  0  . 


(74) 


These  give 

dO            sin  0 

dO        cos  0 

cos  y  -j-  =  .—  , 
eta              ^ 

cos^   ,    =  -.   , 
dy          A 

cfy             sinvy  cos  6 

dy             siny  sin0 

dx                    A 

dy                   A 

dd 
cos  y  -=-  =  0 ; 

c?>?  cosi? 


and  from  (73)  we  get 
dx 

—~—  =  COS  U, 

dr 

dx     _  dx  _ 

dv         dy 


dz 
'dr 


-0: 


£"  =  0> 
dx  . 


dy 
-^-  =  sin«, 

c?y         c?v  dz         dz       A 

-/-  =  -/-  =  r  cos  w,     -  j     - ;  -y-  =  0 ; 
(2v         c?/  dfv        ^/ 


^2 

-^ —  =  —  r  cos  it  SIP  x ; 

ds 

-— -  =  r  sin  w. 
d» 


Substituting  the  values  thus  found,  in  the  equations 

dO  dO     dx    .  dO     dy 

cos  y  3-  =  cos  y  -=-  •  -j — h  cos  7  j~  '  j  > 
dv  ax    av  ay     av 

dy        dy     dx        dy     dy        dy     dz 
dv        dx     dv        dy     dv        dz     d^ 


156  THEORETICAL  ASTRONOMY. 

we  get 

do  do       r        fn 

cos  >?  -=-  =  cos  y  -j-  =  -,  cos  (0  —  u)t 
dv  d%       A 

(76) 


In  a  similar  manner,  we  derive 

do  1    .    fa  d^  1 

cos  *]  —3—  =  —  —  sin  (0  —  u),         —~  =  —  —  sm  y  cos  (0  —  u)t 

dO  df)  r 

cos  V)  -=—  =  0,  --—  =  —  —  cos  yj  sm  i  cos  u,   (77  ) 

do  di)          .  r 

COS  Tj  —  TV-  =  0,  —  yf-  =  -f-  —  COS  I]  Sin  W. 

CM  Cw  d 

If  we  introduce  the  elements  ^,  Jtf"0,  and  //,  which  determine  r  and  v, 
we  have,  from 

de  dO    dr    ,  rf^     c?v 

cos  i?  -j—  =  cos  T?  -j-  •  -j  —  h  cos  i?  -j-  •  -j—  , 
1  d<p  dr    d<f>    [          '  dv    dy 

df)  _  df)     dr        df)     dv 
d<p        dr    dy        dv    d<p' 

if  we  introduce  also  the  auxiliary  quantities  /  and  Fy  as  determined 
by  means  of  the  equations  (59), 

cos  y  -j-  =  ~A  cos  (0  —  u  —  F\     -^-  =  —  tsm7]sm(0  —  u  —  F).    (78) 

Finally,  using  the  auxiliaries  g,  h,  G,  and  H,  according  to  the  equa- 
tions (61),  we  get 

cos  >?  --   =  ?cos(0  —  u—  G), 


,,        ,  i              I                               (79) 

dO        h         ,  dr)              h    .             f                 _. 

cos  r)  — —  =  —  cos  (0  —  u  —  ±L ).  — ^ —  =  —  -j-  sm  TJ  sm  (0  —  u  —  M ). 

d/j-        A  dp             A 

If  we  express  r  and  v  in  terms  of  the  elements  T,  g,  and  e,  the 
values  of  the  auxiliaries  /,  g,  A,  F,  &c.  must  be  found  by  means  of 
(64) ;  and,  in  the  same  manner,  any  other  elements  which  determine 
the  form  of  the  orbit  and  the  position  of  the  body  in  its  orbit,  may 
be  introduced. 

The  partial  differential  coefficients  with  respect  to  the  elements 
having  been  found,  we  have 

do  do  do  dO 

COS  JJ  A0  =  COS  r)  -—-  Ay  4-  COS  f]  -7—  ACP  -f-  COS  r]  -rTT  A Mn  +  COS  Tt  -7-  &fJ. 

d%  dy  dMQ  dp 


DIFFERENTIAL    FORMULA.  157 

from  which  it  appears  that,  by  the  introduction  of  £  as  one  of  the 
elements  of  the  orbit,  when  the  geocentric  places  are  referred  directly 
to  the  plane  of  the  unchanged  orbit  as  the  fundamental  plane,  the 
variation  of  the  geocentric  longitude  in  reference  to  this  plane  depends 
on  only  four  elements. 

57.  It  remains  now  to  derive  the  formulae  for  finding  the  values 
of  T)  and  6  from  those  of  X  and  /?.  Let  XQ,  y0,  z0  be  the  geocentric  co- 
ordinates of  the  body  referred  to  a  system  in  which  the  ecliptic  is 
the  plane  of  xy,  the  positive  axis  of  x  being  directed  to  the  point 
whose  longitude  is  & ;  and  let  #/,  y0',  z0'  be  the  geocentric  co-ordi- 
nates of  the  body  referred  to  a  system  in  which  the  axis  of  x  remains 
the  same,  but  in  which  the  plane  of  the  unchanged  orbit  is  the  plane 
of  xy;  then  we  shall  have 

x0  —  A  cos  /?  cos  (A  —  ft  ),  x0'  =  A  cos  i?  cos  0, 

yo  =  J  cos  /?  sin  (A  —  Q  ),  y0'  =  A  cos  T?  sin  0, 

ZQ  =  J  sin  £,  z0'  =  A  sin  iy, 

and  also 

<  =  *0» 

yo  =  y0  cos  *  -f  zo sin  *» 

z0'  =  —  yQ  sin  i  +  z0  cos  i. 
Hence  we  obtain 

COS  r)  COS  0  =  COS  /?  COS  (A  —  &  ), 

cos  T)  sin  0  =  cos  /?  sin  (A  —  &  )  cos  i  +  sin  £  sin  t,  (80 ) 

sin  17  =  —  cos  £  sin  (A  —  &  )  sin  i  -\-  sin  /?  cos  i. 

These  equations  correspond  to  the  relations  between  the  parts  of  a 
spherical  triangle  of  which  the  sides  are  i,  90°  —  57,  and  90°  —  ft 
the  angles  opposite  to  90°  —  ^  and  90°  —  ft  being  respectively 
90°  -f-  (X  —  ^)  and  90°  —  0.  Let  the  other  angle  of  the  triangle  be 
denoted  by  f,  and  we  have 


cos  17  sin  Y  =  sin  *  cos  (A  —  &  ), 

cos  T?  cos  r  =  sin  i  sin  (A  —  &  )  sin  ft  -f-  cos  i  cos  /?. 


(81) 


The  equations  thus  obtained  enable  us  to  determine  y,  0,  and  f  from 
I  and  /9.  Their  numerical  application  is  facilitated  by  the  intro- 
duction of  auxiliary  angles.  Thus,  if  we  put 


,  , 

n  cos  ^=  cos  j9  sin  (A  —  ft), 


158  THEORETICAL   ASTRONOMY. 

in  which  n  is  always  positive,  we  get 

cos  TI  cos  6  =  cos  /?  cos  (A  —  &  ), 

cos  rj  sin  6  =  n  cos  (N  —  t),  (83) 

sin  7]          =  ra  sin  (JV  —  i\ 

from  which  ^  and  0  may  be  readily  found.     If  we  also  put 

n'  sin  N'  =  cos  i, 
n'  cos  JV  =  sin  i  sin  (A  —  ft  ), 
we  shall  have 

cot  Nr  —  -  tan  i  sin  (A  —  ^  ), 


If  f  is  small,  it  may  be  found  from  the  equation 

sin*  cos  (A  -S» 
cos?? 

The  quadrants  in  which  the  angles  sought  must  be  taken,  are  easily 
determined  by  the  relations  of  the  quantities  involved  ;  and  the 
accuracy  of  the  numerical  calculation  may  be  checked  as  already 
illustrated  for  similar  cases. 

If  we  apply  Gauss's  analogies  to  the  same  spherical  triangle,  we  get 

fiin  (45°  -  j,)  sin  (45°  -  J  (0  +  r))  = 

cos  (45°  -f  i  (A  -  £))  sin  (45°  —  J  (£  +  i)), 
sin  (45°  -  j?)  cos  (45°  -  J  (*  +  r))  = 

sin  (45°  +  i  (A  -  &))  sin  (45°  -  J  (0  —  0), 
cos  (45°  —  J?)  sin  (45°  —  j  (0  —  r))  =  (87) 

cos  (45°  +  i  (A  -  8))  cos  (45°  -  i  (p  +  t)), 
cos  (45°  -  ^)  cos  (45°  -  J  (<?  -  r))  = 

sin  (45°  +  £  C;  -  £))  cos  (45°  -  J  (/?  -  i)), 

from  which  we  may  derive  ^,  0,  and  y. 

When  the  problem  is  to  determine  the  corrections  to  be  applied  to 
the  elements  of  the  orbit  of  a  heavenly  body,  in  order  to  satisfy 
given  observed  places,  it  is  necessary  to  find  the  expressions  for 
cos  7]  A#  and  A^  in  terms  of  cos  /9  AA  and  A/3.  If  we  differentiate  the 
first  and  second  of  equations  (80),  regarding  &  and  i  (which  here 
determine  the  position  of  the  fundamental  plane  adopted)  as  con- 
stant, eliminate  the  terms  containing  dy  from  the  resulting  equations, 
and  reduce  by  means  of  the  relations  of  the  parts  of  the  spheric! 
triangle,  we  get 


NUMERICAL    EXAMPLE.  159 

cos  i)  do  =  cos  Y  cos  p  dl  -f  sin  F  dp. 

Differentiating  the  last  of  equations  (80),  and  reducing,  we  find 
df]  =  —  sin  Y  cos  p  ctt  -f  cos  Y  dp. 

The  equations  thus  derived  give  the  values  of  the  differential  co- 
efficients of  6  and  y  with  respect  to  A  and  /9  ;  and  if  the  differences 
A^  and  A/9  are  small,  we  shall  have 

cos  rj  A0  =  cos  Y  cos  p  A/*  -f  sin  y  A/9, 
A>?  =  —  sin  Y  cos  /5  AA  -f  cos  y  A/1 

The  value  of  p  required  in  the  application  of  numbers  to  these 
equations  may  generally  be  derived  with  sufficient  accuracy  from 
(86),  the  algebraic  sign  of  cosf  being  indicated  by  the  second  of 
equations  (81)  ;  and  the  values  of  rj  and  6  required  in  the  calculation 
of  the  differential  coefficients  of  these  quantities  with  respect  to  the 
elements  of  the  orbit,  need  not  be  determined  with  extreme  accuracy. 

58.  EXAMPLE.  —  Since  the  spherical  co-ordinates  which  are  fur- 
nished directly  by  observation  are  the  right  ascension  and  declina- 
tion, the  formulae  will  be  most  frequently  required  in  the  form  for 
finding  rj  and  0  from  a  and  d.  For  this  purpose,  it  is  only  necessary 
to  write  a  and  d  in  place  of  A  and  /9,  respectively,  and  also  &  ',  V, 
cof,  %',  and  u'  in  place  of  &,  i,  a),  %,  and  u,  in  the  equations  which 
have  been  derived  for  the  determination  of  rj  and  0,  and  for  the 
differential  coefficients  of  these  quantities  with  respect  to  the  elements 
of  the  orbit. 

To  illustrate  this  clearly,  let  it  be  required  to  find  the  expressions 
for  cos  rj  A#  and  &rj  in  terms  of  the  variations  of  the  elements  in  the 
case  of  the  example  already  given  ;  for  which  we  have 

<//  =  50°  10'  7".29,        &'  =  353°  45'  35".87,        i'  =  19°  26'  25".76. 

These  are  the  elements  which  determine  the  position  of  the  orbit  of 
Evsrynome  @,  referred  to  the  mean  equinox  and  equator  of  1865.0. 
We  have,  further, 


log/=  0.62946,  log#  =  0.34593,  log  h  =  2.97759, 

F=  339°  14'  0",  G  =  350°  11'  16",         H=  14°  30'  48';, 

u'  =  179°  13'  58". 

In  the  first  place,  we  compute  ^,  0,  and  7-  by  means  of  the  formulae 


160  THEORETICAL   ASTRONOMY. 

(83)  and  (85),  or  by  means  of  (87),  writing  a,  d,  &',  and  V  instead 
of  ^,  ft,  &,  and  i9  respectively.  Hence  we  obtain 

e  =  188°  31'  9",  T]  =  —  1°  59'  28",  r  =  — 19°  17'  7". 

Since  the  equator  is  here  considered  as  the  fundamental  plane,  the 
longitude  6  is  measured  on  the  equator  from  the  place  of  the  ascend- 
ing node  of  the  orbit  on  this  plane.  The  values  of  the  differential 
Coefficients  are  then  found  by  means  of  the  formulae 

dO  di)  r 

cos  f)  ^—7  =  0,  j—r  =  —  -  cos  T)  sm  j  cos  tr, 

de  dy          ,  r 

cos  TI  -^r  =  0,  -jj-  =  4  -  cos  rj  sm  u , 

cos  7]  --r  =  T-  cos  (0  —  u')t  --  =  —     sin  T?  sin  (0  —  if), 


cos 7]  ,,,  =  —  cos  (e  —  u'  —  G\  ,,,  =  —  ^-  sin -n  sin  (0  —  u'  —  GO, 

dM0       A  dM0            A 

Mhc^(0-u'-H\  ^L  =  -^s[n7ism(e-u'-H)t 

dfj.        A        ^  dfj.             A        '                            '* 


which  give 


008,^  =  0,  ^  =  +  0.5072, 

^  =  0,  A.  =  +  0.0204, 


di!  di' 

-J-  =  + 1.5051,  -|r  =  +  0- 

-^-  =  +  2.0978,  -^-  =  +  0.0422. 


dff  =-1-1.1922,  -^  =  +  0.0143, 

ClMn 


~  =  +  538.00,  -^?-= -1.71. 


Therefore,  the  equations  for  cos  7]  &6  and  &y  become 

cos  ^  A0  =  4  1.5051  A/  4  2.0978  A?  4  1.1922  Ajtf0  4  538.00 
AT?  =  4  0.0086  A/  4  0.0422  A?.  4  0.0143  A Jf0  —  1.71  A/* 
4  0.5072  A ^'4  0.0204  Ai'. 

If  we  assign  to  the  elements  of  the  orbit  the  variations 


DIFFERENTIAL   FORMULA.  161 

AO/=  -6".64,  A&'  =  — 14".12,  AI*  =  —  8".86, 

A?  ==  -f  10",  Aj/0  =  -|-  10",  A//  =  -f  0".01, 

we  have 

A/  =  A«/  +  cos  i'  A&'  =  —  19".96 ; 

and  the  preceding  equations  give 

cos  y  A0  =  +  8".24,  AT;  =  —  6".96. 

With  the  same  values  of  AO/,  A  & ',  &c.,  we  have  already  found 

cos  8  Aa  =  +  5".47,  Ad  ==  —  9".29, 

which,  by  means  of  the  equations  (88),  writing  a  and  d  in  place  of 
A  and  /9,  give 

cos  ti  A0  =  -f  8".23,  AI?  =  —  6".96. 

59.  In  special  cases,  in  which  the  differences  between  the  calcu- 
lated and  the  observed  values  of  two  spherical  co-ordinates  are  given, 
and  the  corrections  to  be  applied  to  the  assumed  elements  are  sought, 
it  may  become  necessary,  on  account  of  difficulties  to  be  encountered 
in  the  solution  of  the  equations  of  condition,  to  introduce  other  ele- 
ments of  the  orbit  of  the  body.  The  relation  of  the  elements  chosen 
to  those  commonly  used  will  serve,  without  presenting  any  difficulty, 
for  the  transformation  of  the  equations  into  a  form  adapted  to  the 
special  case.  Thus,  in  the  case  of  the  elements  which  determine  the 
form  of  the  orbit,  we  may  use  a  or  log  a  instead  of  ft,  and  the 
equation 

kVl  -f  m 

"=   -IT 

gives 

dft  =  —  j-efo  =  — jyefloga,  (89) 

a  /0 

in  which  ^0  is  the  modulus  of  the  system  of  logarithms.  Therefore, 
the  coefficient  of  A//  is  transformed  into  that  of  A  log  a  by  multiply- 
ing it  by  —  f  £;  and  if  the  unit  of  the  rath  decimal  place  of  the  loga- 

A 

rithms  is  taken  as  the  unit  of  A  loga,  the  coefficient  must  be  also 
multiplied  by  10~m.  The  homogeneity  of  the  equation  is  not  disturbed, 
since  /J.  is  here  supposed  to  be  expressed  in  seconds. 

If  we  introduce  logjp  as  one  of  the  elements,  from  the  equation 

p  =  a  cos*  <p 
11 


162  THEORETICAL   ASTRONOMY. 

we  get 

d  logp  =  —  I  -  dfj.  —  2/*0  tan  <p  d<p, 
or 

dfj.  =  —  |  y-  d  logp  —  3/A  tan  <f>  d<p.  (90) 

0 

Hence  it  appears  that  the  coefficients  of  A  logp  are  the  same  as  those 
of  A  log  a,  but  since  p  is  also  a  function  of  <py  the  coefficients  of  &<p 

are  changed;  and  if  we  denote  by  cos  d  I  -7—  I  and  I  -7—  I  the  values  of 

the  partial  differential  coefficients  when  the  element  fj.  is  used  in  con- 
nection with  (p,  we  shall  have,  for  the  case  under  consideration, 

.  da  „  /  da  \        np.  _.  da, 

cos  o  -7—  =  cos  o    -7—  I  —  3  -  tan  y  cos  o  -y-, 
d(f>  \  d</>  i         s  a/j. 

dd  I  dd\       nfi  dd 


in  which  s  =  206264".8.     If  the  values  of  the  differential  coefficients 
with  respect  to  fjt  and  <p  have  not  already  been  found,  it  will  be  ad- 

dr    dv        dr  dv       . 

vantageous  to  compute  the  values  ot  -r->  -7—  »  ~n  -  >  and  -7-=  -  by 

d<p    d<p    d  iogp  d  iogp 

means  of  the  expressions  which  may  be  derived  by  substituting  in 
the  equations  (15)  the  value  of  dfj.  given  by  (90),  and  then  we  may 

compute  directly  the  values  of  cos  d-j—,  cos  d  -^  -  ,  -j—  ,  and  -ji  --- 

dtp  d  logjt?   d(p  d  logp 

In  place  of  Mw  it  is  often  convenient  to  introduce  L0,  the  mean 
longitude  for  the  epoch  ;  and  since 

V=*+«i 

we  have 

dL0  =  dM0  +d7t  =  dMQ  +  da»  +  d  ft, 

and,  when  £  is  used, 

d:/  +  (1  —  cos  i)  rfft  . 


Instead  of  the  elements  ft  and  i  which  indicate  the  position  of  the 
plane  of  the  orbit,  we  may  use 

b  =  sin  i  sin  ft,  c  =  sin  i  cos  ft, 

and  the  expressions  for  the  relations  between  the  differentials  of  b 
and  c  and  those  of  i  and  ft  are  easily  derived.  The  cosines  of  the 
angles  which  the  line  of  apsides  or  any  other  line  in  the  orbit  makes 
with  the  three  co-ordinate  axes,  may  also  be  taken  as  elements  of  the 


DIFFEBENTIAL   FORMULAE.  163 

orbit  in  the  formation  of  the  equations  for  the  variation  of  the  geo- 
centric place. 

60.  The  equations  (48),  by  writing  I  and  b  in  place  of  ^  and  /S, 
respectively,  will  give  the  values  of  the  differential  coefficients  of 
the  heliocentric  longitude  and  latitude  with  respect  to  #,  y,  and  z. 
Combining  these  with  the  expressions  for  the  differential  coefficients 
of  the  heliocentric  co-ordinates  with  respect  to  the  elements  of  the 
orbit,  we  obtain  the  values  of  cos  b  A/  and  A&  in  terms  of  the  varia- 
tions of  the  elements. 

The  equations  for  dx}  dy,  and  dz  in  terms  of  du,  dft,  and  di,  may 
also  be  used  to  determine  the  corrections  to  be  applied  to  the  co-or- 
dinates in  order  to  reduce  them  from  the  ecliptic  and  mean  equinox 
of  one  epoch  to  those  of  another,  or  to  the  apparent  equinox  of  the 
date.  In  this  case,  we  have 

du  =  dn  —  d  &  . 

When  the  auxiliary  constants  A,  B,  a,  6,  &c.  are  introduced,  to 
find  the  variations  of  these  arising  from  the  variations  assigned  to 
the  elements,  we  have,  from  the  equations  (99)1? 

cot  A  =  —  tan  ft  cos  i, 

cot  B  =  cot  &  cos  i  —  sin  i  cosec  ft  tan  e, 

cot  C  =  cot  ft  cos  i  -f-  sin  i  cosec  ft  cot  e, 

in  which  i  may  have  any  value  from  0°  to  180°.  If  we  differentiate 
these,  regarding  all  the  quantities  involved  as  variable,  and  reduce 
by  means  of  the  values  of  sin  a,  sin  6,  and  sin  c,  we  get 

,  .        cos  i    ,  sin  A    .          .    .  ,. 

dA  =    .        rfft  --  :  -  sin  ft  sm  i  di, 
sm2a  sma 


dB  =    .  2     (cos  i  cos  e  —  sin  i  sin  e  cos  ft  )  c?ft 
sin  b 


.  .  .    .  •  •     \  j-  i 

-|  —  -.  —  7-  (cos  &  smi  cos  e  -j-  cos  i  sin  e)  di  -| 


-.    7 

dC=    .  2     (cos  i  sin  e  -{-  sm  i  cos  e  cos  &)  d& 
sin  c 

,    sin  C  ,  ...  .         .   ..      sinisin 

H  —  ;  -  (cos  ft  SHU  sm  e  —  cos  i  cos  e)  di  H  --  —„ 
sine  sm*c 

and  these,  by  means  of  (101)^  reduce  to 


164  THEORETICAL   ASTRONOMY. 


7  .        cos  i    7  ^ 

dA  =    .        dQ  —  sin  A  cot  a  di, 
2 


cos  e  cos  e  ,  .    ^        .   .  .       cos  a   , 

.cfe,  (91) 


,~  sinecosi  7_         .    ~          T  ,    cos  a    , 

dC=  --  :—=  —  a&  —  sm  C  cot  c  (fo  -f-    .  9    cfe. 
sin2  c  '    sm*  c 

Let  us  now  differentiate  the  equations  (101)w  using  only  the  upper 
sign,  and  the  result  is 

da  =  —  sin  i  sin  A  dQ  -f  cos  A  di, 

db  =  —  sin  i  sin  B  d&  -f  cos  Bdi-\-  cos  c  cosec  b  de, 

de  =  —  sin  i  sin  C  dQ  -f-  cos  C  di  —  cos  b  cosec  c  de. 

If  we  multiply  the  first  of  these  equations  by  cot  a,  the  second  by 
cot  6,  and  the  third  by  cot  c,  and  denote  by  ^0  the  modulus  of  the 
system  of  logarithms,  we  get 

d  log  sin  a  =  —  A0  sin  i  cot  a  sin  A  d  &  -f  ^o  cot  a  cos  ^  di, 

d  log  sin  6  =  —  ^  sin  i  cot  b  sin  2?  c?&  -f  J0  cot  6  cos  I»  di  -f  ^0  S 


1.  ...  ~   -.  _,  /-v7' 

log  sin  c=  —  A0smi  cote  sin  CdQ  -\-XQ  cote  cos  Cdi  — 


~ 
sin  o 

COS  &  COS  C 


a  . 

sm  c 

(92) 

The  equations  (91)  and  (92)  furnish  the  differential  coefficients  of 
At  B,  Cj  log  sin  a,  &c.  with  respect  to  &,  i,  and  e;  and  if  the  varia- 
tions assigned  to  &,  i,  and  e  are  so  small  that  their  squares  may  be 
neglected,  the  same  equations,  writing  &A,  A&,  At,  &c.  instead  of 
the  differentials,  give  the  variations  of  the  auxiliary  constants.  In 
the  case  of  equations  (92),  if  the  variations  of  &,  i,  and  e  are  ex- 
pressed in  seconds,  each  term  of  the  second  member  must  be  divided 
by  206264.8,  and  if  the  variations  of  log  sin  a,  log  sin  6,  and  log  sine 
are  required  in  units  of  the  mth  decimal  place  of  the  logarithms,  each 
term  of  the  second  member  must  also  be  divided  by  10™. 

If  we  differentiate  the  equations  (81)D  and  reduce  by  means  of  the 
same  equations,  we  easily  find 

cos  b  dl  =  cos  i  sec  b  du  -\-  cos  b  d  Q  —  sin  b  cos  (I  —  &  )  di,     /QO  \ 
db  =  sin  i  cos  (I  —  Q>  )  du  -f-  sin  (I  —  Q>  )  di, 

which  determine  the  relations  between  the  variations  of  the  elements 
of  the  orbit  and  those  of  the  heliocentric  longitude  and  latitude. 
By  differentiating  the  equations  (88)w  neglecting  the  latitude  of 


DIFFERENTIAL   FORMULA.  165 

the  sun,  and  considering  1,  /9,  J,  and  O  as  variables,  we  derive,  after 
reduction, 

r> 

cos  /?  dX  =  _  cos  (A  —  O)dO, 

R  (94) 

dp  =  —  --  sin  £  sin  A  —  O 


which  determine  the  variation  of  the  geocentric  latitude  and  longitude 
arising  from  an  increment  assigned  to  the  longitude  of  the  sun.  It 
appears,  therefore,  that  an  error  in  the  longitude  of  the  sun  will 
produce  the  greatest  error  in  the  computed  geocentric  longitude  of  a 
heavenly  body  when  the  body  is  in  opposition. 


166  THEORETICAL   ASTRONOMY. 


CHAPTER   III. 

INVESTIGATION  OP  FORMULAE  FOR  COMPUTING  THE  ORBIT  OF  A  COMET  MOVING  IN 
A  PARABOLA,  AND  FOR  CORRECTING  APPROXIMATE  ELEMENTS  BY  THE  VARIATION 
OF  THE  GEOCENTRIC  DISTANCE. 

61.  THE  observed  spherical  co-ordinates  of  the  place  of  a  heavenly 
body  furnish  each  one  equation  of  condition  for  the  correction  of  the 
elements  of  its  orbit  approximately  known,  and  similarly  for  the 
determination  of  the  elements  in  the  case  of  an  orbit  wholly  unknown ; 
and  since  there  are  six  elements,  neglecting  the  mass, — which  must 
alwrays  be  done  in  the  first  approximation,  the  perturbations  not 
being  considered, — three  complete  observations  will  furnish  the  six 
equations  necessary  for  finding  these  unknown  quantities.  Hence, 
the  data  required  for  the  determination  of  the  orbit  of  a  heavenly 
body  are  three  complete  observations,  namely,  three  observed  longi- 
tudes and  the  corresponding  latitudes,  or  any  other  spherical  co- 
ordinates which  completely  determine  three  places  of  the  body  as 
seen  from  the  earth.  Since  these  observations  are  given  as  made  at 
some  point  or  at  different  points  on  the  earth's  surface,  it  becomes 
necessary  in  the  first  place  to  apply  the  corrections  for  parallax.  In 
the  case  of  a  body  whose  orbit  is  wholly  unknown,  it  i?  impossible 
to  apply  the  correction  for  parallax  directly  to  the  place  of  the  body ; 
but  an  equivalent  correction  may  be  applied  to  the  places  of  the 
earth,  according  to  the  formula?  which  will  be  given  in  the  next 
chapter.  However,  in  the  first  determination  of  approximate  ele- 
ments of  the  orbit  of  a  comet,  it  will  be  sufficient  to  neglect  entirely 
the  correction  for  parallax.  The  uncertainty  of  the  observed  places 
of  these  bodies  is  so  much  greater  than  in  the  case  of  well-defined 
objects  like  the  planets,  and  the  intervals  between  the  observations 
which  will  be  generally  employed  in  the  first  determination  of  the 
orbit  will  be  so  small,  that  an  attempt  to  represent  the  observed  places 
with  extreme  accuracy  will  be  superfluous. 

When  approximate  elements  have  been  derived,  we  may  find  the 
distances  of  the  comet  from  the  earth  corresponding  to  the  three 
observed  places,  and  hence  determine  the  parallax  in  right  ascension 


DETERMINATION   OF   AN   ORBIT.  167 

and  in  declination  for  each  observation  by  means  of  the  usual  formulae. 
Thus,  we  have 

?r/>  cos  <f>'    sin  (o  —  0) 
A  cos  d 

tanr  = 


COS  (a  —  0)' 

sin  <p'    sin  (p  — 


Ad  = 


in  which  a  is  the  right  ascension,  d  the  declination,  A  the  distance 
of  the  comet  from  the  earth,  tpf  the  geocentric  latitude  of  the  place 
of  observation,  0  the  sidereal  time  corresponding  to  the  time  of 
observation,  p  the  radius  of  the  earth  expressed  in  parts  of  the 
equatorial  radius,  and  n  the  equatorial  horizontal  parallax  of  the 
sun. 

In  order  to  obtain  the  most  accurate  representation  of  the  observed 
place  by  means  of  the  elements  computed,  the  correction  for  aberra- 
tion must  also  be  applied.  When  the  distance  A  is  known,  the 
time  of  observation  may  be  corrected  for  the  time  of  aberration; 
but  if  A  is  not  approximately  known,  this  correction  may  be  neglected 
in  the  first  approximation. 

The  transformation  of  the  observed  right  ascension  and  declination 
into  latitude  and  longitude  is  effected  by  means  of  the  equations 
which  may  be  derived  from  (92)x  by  interchanging  a  and  ^,  d  and  ft, 
and  writing  —  e  instead  of  e.  Thus,  we  have 

AT      tan  d 

tan  N=  -  —  , 

sin  a 

,         cos(J!V"  —  e)  ,.,. 

tan  A  =  -  -  —  =r=^tana,  (1) 

cosN 

tan  /5  =  tan  (N  —  e  )  sin  A, 
and  also 

cos  (N  —  s)  _  cos  /?  sin  A 
cos  N        ~  cos  <5  sin  a' 

which  will  serve  to  check  the  numerical  calculation  of  ^  and  /?. 
Since  cos  /9  and  cos  d  are  always  positive,  cos  A  and  cos  a  must  have 
the  same  sign,  thus  determining  the  quadrant  in  which  A  is  to  be 
taken. 

62.  As  soon  as  these  preliminary  corrections  and  transformations 
have  been  effected,  and  the  times  of  observation  have  been  reduced 
to  the  same  meridian,  the  longitudes  having  been  reduced  to  the 


168  THEORETICAL   ASTRONOMY. 

same  equinox,  we  are  prepared  to  proceed  with  the  determination  of 
the  elements  of  the  orbit.  For  this  purpose,  let  t,  tr,  t"  be  the  times 
of  observation,  r,  r',  r"  the  radii-vectores  of  the  body,  and  u,  u',  u" 
the  corresponding  arguments  of  the  latitude,  jR,  Rr,  R"  the  distances 
of  the  earth  from  the  sun,  and  O,  O',  O"  the  longitudes  of  the  sun 
corresponding  to  these  times. 

Let  [rr;]  denote  double  the  area  of  the  triangle  formed  between 
the  radii-vectores  r,  r'  and  the  chord  of  the  orbit  between  the  corre- 
sponding places  of  the  body,  and  similarly  for  the  other  triangles 
thus  formed.  The  angle  at  the  sun  in  this  triangle  is  the  difference 
between  the  corresponding  arguments  of  the  latitude,  and  we  shall 

have 

[rr'~\    —  rr'  sin  (ur  —  u}, 

[rr"]  =rr"sm(u"  —  u),  (2) 

[rV']=ry'sin(ti"--tt'), 

If  we  designate  by  x,  y,  z,  a?',  y',  zr,  x",  y",  z"  the  heliocentric  co- 
ordinates of  the  body  at  the  times  t,  tr,  and  t",  we  shall  have 

x  =  r  sin  a  sin  (A  -f-  it), 
x'  =  r'  sin  a  sin  ( A  -f-  u'\ 


in  which  a  and  A  are  auxiliary  constants  which  are  functions  of  the 
elements  Q>  and  i,  and  these  elements  may  refer  to  any  fundamental 
plane  whatever.  If  we  multiply  the  first  of  these  equations  by 
sin  (it"  —  it'),  the  second  by  —  sin  (u"  —  it),  and  the  third  by 
sin  (uf  —  it),  and  add  the  products,  we  find,  after  reduction, 

-  sin  (u"  —  u'}  —  ~  sin  (u"  —  u)  -f  ^  sin  (u'  —  u)  =  0, 

which,  by  introducing  the  values  of  [rr'],  [rr"],  and  [r'r"],  becomes 

[//']  x  —  [rr"]  x'  +  [rr']  x"  =  0. 
If  we  put 

[r'r"]  „      [rr']  ,Q, 

n==D^J  n      FT 

weget  >      " "_  r^ 

In  precisely  the  same  manner,  we  find 

ny  —  */'  +  n"f  =  0,  _ 

>'    i   ~"~"__n  W 


DETERMINATION   OF   AN   ORBIT.  169 

~ince  the  coefficients  in  these  equations  are  independent  of  the  posi- 
tions of  the  co-ordinate  planes,  except  that  the  origin  is  at  the  centre 
of  the  sun,  it  is  evident  that  the  three  equations  are  identical,  ana 
express  simply  the  condition  that  the  plane  of  the  orbit  passes  through 
the  centre  of  the  sun  ;  and  the  last  two  might  have  been  derived 
from  the  first  by  writing  successively  y  and  z  in  place  of  x. 

Let  ^,  A',  X"  be  the  three  observed  longitudes,  /9,  /?',  ft"  the  corre- 
sponding latitudes,  and  A,  J',  A"  the  distances  of  the  body  from  the 
earth  ;  and  let 

A  cos  p  =  P,  A'  cos  p  =  f>',  A"  cos  p'  =  //', 

which  are  called  curtate  distances.     Then  we  shall  have 

x  =  p  cos  A  —  R  cos  O  ,  d  =  p'  cos  A'  —  R'  cos  Q  ', 

y  =  p  sin  A  —  R  sin  O  ,  y'  =  p'  sin  X'  —  R'  sin  O', 

z  —  p  tan  ft,  z'  =  p'  tan  p, 

of'  =  P"  cos  I"  —  .fl"cosQ", 


in  which  the  latitude  of  the  sun  is  neglected.  The  data  may  be  so 
transformed  that  the  latitude  of  the  sun  becomes  0,  as  will  be  ex- 
plained in  the  next  chapter  ;  but  in  the  computation  of  the  orbit  of 
a  comet,  in  which  this  preliminary  reduction  has  not  been  made,  it 
will  be  unnecessary  to  consider  this  latitude  which  never  exceeds  1", 
while  its  introduction  into  the  formulae  would  unnecessarily  com- 
plicate some  of  those  which  will  be  derived.  If  we  substitute  these 
values  of  x,  x>  ',  &c.  in  the  equations  (4)  and  (5),  they  become 

0  =  n  (p  cos  A  —  R  cos  O  )  —  (pr  cos  /  —  Rr  cos  O  ') 

+  n"  (p"  cos  A"  —  R"  cos  O"), 
0  =  n  (p  sin  A  —  R  sin  O  )  —  (p  sin  X'  —  R'  sin  O')  (6) 

+  n"  (p"  sin  A"  —  R"  sin  0  "), 
0  =  np  tan  /9  —  p'  tan  p  -f  ri'p"  tan  ft". 

These  equations  simply  satisfy  the  condition  that  the  plane  of  the 
orbit  passes  through  the  centre  of  the  sun,  and  they  only  become 
distinct  or  independent  of  each  other  when  n  and  n"  are  expressed 
in  functions  of  the  time,  so  as  to  satisfy  the  conditions  of  undisturbed 
motion  in  accordance  with  the  law  of  gravitation.  Further,  they 
involve  five  unknown  quantities  in  the  case  of  an  orbit  wholly 
unknown,  namely,  w,  n",  py  p',  and  p"  ;  and  if  the  values  of  n  and 
n"  are  first  found,  they  will  be  sufficient  to  determine  p,  p',  and  p". 


170  THEOEETICAL   ASTRONOMY. 

The  determination,  however,  of  n  and  n"  to  a  sufficient  degree  of 
accuracy,  by  means  of  the  intervals  of  time  between  the  obsei  vations, 
requires  that  pf  should  be  approximately  known,  and  hence,  in 
general,  it  will  become  necessary  to  derive  first  the  values  of  n,  n"y 
and  pr  •  after  which  those  of  p  and  ptf  may  be  found  from  equations 
(6)  by  elimination.  But  since  the  number  of  equations  will  then 
exceed  the  number  of  unknown  quantities,  we  may  combine  them  in 
such  a  manner  as  will  diminish,  in  the  greatest  degree  possible,  tho 
effect  of  the  errors  of  the  observations.  In  special  cases  in  which 
the  conditions  of  the  problem  are  such  that  when  the  ratio  of  two 
curtate  distances  is  known,  the  distances  themselves  may  be  deter- 
mined, the  elimination  must  be  so  performed  as  to  give  this  ratio 
with  the  greatest  accuracy  practicable. 

63.  If,  in  the  first  and  second  of  equations  (6),  we  change  the 
direction  of  the  axis  of  x  from  the  vernal  equinox  to  the  place  of  the 
sun  at  the  time  t't  and  again  in  the  second,  from  the  equinox  to  the 
second  place  of  the  body,  we  must  diminish  the  longitudes  in  these 
equations  by  the  angle  through  which  the  axis  of  x  has  been  moved, 
and  we  shall  have 

0  =  n(p  cos(A  —  0')  — jRcos(O'—  O))  —  0'  cos  (A'  —  0')  —  #) 
+  n"  (P"  cos  (A"_  00  -  R"  cos  (O"  -  00), 

0  =  7i  0>  sin  (A  —  0')+jRsm(0'—  O))—  /sin(A'  —  0') 

+  n"  (,"  sin  (A"  _  00  -R"  Bin(0"-  ©')),  (7) 

0  =  nO>sin(A'  —  A)  +jRsin(0  —  /))  —  R  sin(O'—  A') 
-  n"  (p"  sin  (A"  —  A')  —  R"  sin  (O  "  —  *')), 

0  =  np  tan  p  —  p'  tan  p  -f  ri'p"  tan  p'. 

If  we  multiply  the  second  of  these  equations  by  tan/3',  and  the 
fourth  by  —  sin  (A' —  0'),  and  add  the  products,  we  get 

0  =  n"P"  (tan  p'  sin  (A"  —  O  0  —  tan  0"  sin  (A'  —  O  0) 
—  n".R"sin(0"—  0  0  tan  jfiH- n/»  (tan  ^  sin  (A  •—  0')—  tan/?sin(A'— OO) 
+  nJ2sin(©'— 0  )tan/3/.  (8) 

Let  us  now  denote  double  the  area  of  the  triangle  formed  by  the 
sun  and  two  places  of  the  earth  corresponding  to  R  and  Rf  fry 
and  we  shall  have 

[£#]=  KRfsm(Q'  —  O), 
and  similarly 

[ JH2"]  =  RR"  sin  (  O  "  —  O  ), 
']  =  R'R"  sin(O"—  00- 


ORBIT  OF  A  HEAVENLY  BODY.  17l 

Then,  if  we  put 


_  , 

[£#']'  ~  IEK"Y 

we  obtain 


Substituting  this  in  the  equation  (8),  and  dividing  by  the  coefficient 
of  p",  the  result  is 


_  Q_ 

~  p  n"  '  tan  ft"  sin  (A'  —  Q  ')  —  tan  ft'  sin  (A"  —  Q  ') 


V       tan  ft'  sin  (A  —  Q  ')  —  tan  ft  sin  (A'  —  Q  ') 
n  (A'  —  Q  ')  —  tan  ft'  sin  (A"  —  Q  ' 
_!L    _^M  _  J?sin(0'—  Q)tanff'  _ 
n"       N"  J  tan  /3"  sin  (A'  -  Q')  —  tan  /3'  sin  (A"  —  0')' 

Let  us  also  put 

M'  =  tan  ^  sin  (;  ~  Q  ')  ~  tan  ^  sin  (*'  —  Q') 
~~  tan  ft"  sin  (A'  —  0')  —  tan  ft'  sin  (A"  —  0')' 

j^  „  __  _  sin(0'-0)tanf  __ 
tan/5"  sin  (A'—  0')  —  tan/3'  sin  (A"—  0')' 

and  the  preceding  equation  reduces  to 

-*,  JTR  (11) 


"We  may  transform  the  values  of  3/r  and  M  "  so  as  to  be  better 
adapted  to  logarithmic  calculation  with  the  ordinary  tables.  Thus, 
if  w'  denotes  the  inclination  to  the  ecliptic  of  a  great  circle  passing 
through  the  second  place  of  the  comet  and  the  second  place  of  the 
sun,  the  longitude  of  its  ascending  node  will  be  O',  and  we  shall 

have 

sin  (A'  —  O')  tan  w'  =  tan  ft'.  (12) 

Let  /90,  ftQ"  be  the  latitudes  of  the  points  of  this  circle  corresponding 
to  the  longitudes  A  and  A",  and  we  have,  also, 

tan  ft0  =  sin  (A  —  0')  tan  w', 
tan  ft"  =  sin  (A"  —  Q')  tan  w'. 

Substituting  these  values  for  tan/9',  sin  (A—  O7)  and  sin  (A"—  O;) 
in  the  expressions  for  M  '  and  M  ",  and  reducing,  they  become 

sin(/y-  /?)      cos  p'  cos  ft0" 
sin  (ft"  —  /30")  '    cos  ft0  cos  ft  ' 


=^  Sin(0'- 


172  THEORETICAL  ASTRONOMY. 

When  the  value  of  -77  has  been  found,  equation  (11)  will  give  the 

relation  between  p  and  p"  in  terms  of  known  quantities.  It  is  evi- 
dent, however,  from  equations  (14),  that  when  the  apparent  path  of 
the  comet  is  in  a  plane  passing  through  the  second  place  of  the 

sun,  since,  in  this  case,  ft  —  /?0  and  ft"  =  /?0",  we  shall  have  M'=  ^ 

and  Mfr  =  oo.  In  this  case,  theiefore,  and  also  when  /90  —  ft  and 
ft" — ftQ"  are  very  nearly  0,  we  must  have  recourse  to  some  other 
equation  which  may  be  derived  from  the  equations  (7),  and  which 
does  not  involve  this  indetermination. 

It  will  be  observed,  also,  that  if,  at  the  time  of  the  middle  obser- 
vation, the  comet  is  in  opposition  or  conjunction  with  the  sun,  the 
values  of  Mf  and  M "  as  given  by  equation  (14)  will  be  indeter- 
minate in  form,  but  that  the  original  equations  (10)  will  give  the 
values  of  these  quantities  provided  that  the  apparent  path  of  the 
comet  is  not  in  a  great  circle  passing  through  the  second  place  of  the 
sun.  These  values  are 

,_        sin  (A—  0')  _        sin(Q'—  Q) 

"  sin  (A"—©')'  "  sm(A"_  0') ' 

Hence  it  appears  that  whenever  the  apparent  path  of  the  body  is 
nearly  in  a  plane  passing  through  the  place  of  the  sun  at  the  time  of 
the  middle  observation,  the  errors  of  observation  will  have  great 
influence  in  vitiating  the  resulting  values  of  Mf  and  M"  •  and  to 
obviate  the  difficulties  thus  encountered,  we  obtain  from  the  third  of 
equations  (7)  the  following  value  of  p" : — 


„  _      n      sin  (A'  —  A) 
p  ==p  f 


^E  sin  (G  -  A')  -  ^  #  sin  (©'  -  AO  + 12"  sin  (©"  -  X) 


(15) 


We  may  also  eliminate  p  between  the  first  and  fourth  of  equa- 
tions (7).  If  we  multiply  the  first  by  tan/3',  and  the  second  by 
—  cos (A'—  ©'),  and  add  the  products,  we  obtain 

0  =  n"p"  (tan  f  cos  (A"  —  0')  —  tan  /?"  cos  (A'  —  ©')) 

— ri'R"  tan  p  cos(O"—  0')  +  w/>  (tan /3' cos  (A  —  Q')  —  tan/?cos(A'— 00) 

—  nR  tan  fi  cos  (O'  —  O)  +  R'  tan  /3', 

from  which  we  derive 


ORBIT   OF   A   HEAVENLY   BODY.  173 

"_  0JL     tan  ^  cos  (A  —  Q  ')  —  tan  ff  cos  (A'  —  Q') 
p    ~~P  n"  '  tan  /?"  cos  (A'  —  0')  —  tan  /5'  cos  (A"  —  Q  ')  (16) 

R"  tan/?'  cos(O"—  00  +  —7  12  tan  ^  COB  (0'—  0)  —  —•  R'tan/? 
tan  /S"  cos  (A'  —  Q  ')  —  tan  $  cos  (A"  —  Q') 

Let  us  now  denote  by  I'  the  inclination  to  the  ecliptic  of  a  great 
circle  passing  through  the  second  place  of  the  comet  and  that  point 
of  the  ecliptic  whose  longitude  is  0'  —  90°,  which  will  therefore  be 
the  longitude  of  its  ascending  node,  and  we  shall  have 

cos  (A'  —  0')  tan  I'  =  tan  p  ;  (17) 

and,  if  we  designate  by  /9,  and  /?„  the  latitudes  of  the  points  of  this 
circle  corresponding  to  the  longitudes  ^  and  X",  we  shall  also  have 

tan  /?,  =  cos  (A  —  0')tan.T', 
tan  /?„  =  cos  (A"  —  0')  tan  I'. 

Introducing  these  values  into  equation  (16),  it  reduces  to 

„  _       n       sin  (/9,  —  /9)      cos  /3"  cos  /?„ 
~  p        ' 


sin  (/5"  —  /?„)     cos  /5  cos  /?,  (19) 

tan  r  cos  /5"  cos/?,,  /     „        f     „         ,.    .    n  f     .  R'  \ 

rinC9"-A()       (^"^(e'-GO  +  ^^cosCG'-Q)-^). 

from  which  it  appears  that  this  equation  becomes  indeterminate  when 
the  apparent  path  of  the  body  is  in  a  plane  passing  through  that 
point  of  the  ecliptic  whose  longitude  is  equal  to  the  longitude  of  the 
second  place  of  the  sun  diminished  by  90°.  In  this  case  we  may  use 
equation  (11)  provided  that  the  path  of  the  comet  is  not  nearly  in 
the  ecliptic.  When  the  comet,  at  the  time  of  the  second  observation, 
is  in  quadrature  with  the  sun,  equation  (19)  becomes  indeterminate 
in  form,  and  we  must  have  recourse  to  the  original  equation  (16), 
which  does  not  necessarily  fail  in  this  case. 

When  both  equations  (11)  and  (16)  are  simultaneously  nearly  in- 
determinate, so  as  to  be  greatly  affected  by  errors  of  observation,  the 
relation  between  p  and  p"  must  be  determined  by  means  of  equation 
(15),  which  fails  only  when  the  motion  of  the  comet  in  longitude  is 
very  small.  It  will  rarely  happen  that  all  three  equations,  (14), 
(15),  and  (16),  are  inapplicable,  and  when  such  a  case  does  occur  it 
will  indicate  that  the  data  are  not  sufficient  for  the  determination  of 
the  elements  of  the  orbit.  In  general,  equation  (16)  or  (19)  is  to  be 
used  when  the  motion  of  the  comet  in  latitude  is  considerable,  and 
equation  (15)  when  the  motion  in  longitude  is  greater  than  in  latitude. 


174  THEORETICAL   ASTRONOMY. 

64.  The  formulae  already  derived  are  sufficient  to  determine  the 
relation  between  //'  and  p  when  the  values  of  n  and  n"  are  known, 
and  it  remains,  therefore,  to  derive  the  expressions  for  these  quan- 
tities. 

If  we  put 

t(f   -f)  =  r", 

k  («"  -  O  =  r,  (20) 


and  express  the  values  of  x,  y,  z,  xlfy  y",  z"  in  terms  of  re',  yf,  zr  by 
expansion  into  series,  we  have 

f_dx^    T"      J_    dV    r^_  __  1_    dV    r"8 
x  ~<ft  '*  +1.2*  <ft«*  #  ~    1.2.3*  <#*#  H        ' 

„_  <fo'     r  1      dV      r2  1        #a!      r» 

h"^"*T  +  L2"^'^"  +  r^3'^'^~ 

and  similar  expressions  for  y,  y",  z,  and  zff.  We  shall,  however,  take 
the  plane  of  the  orbit  as  the  fundamental  plane,  in  which  case  2,  z', 
and  z"  vanish. 

The  fundamental  equations  for  the  motion  of  a  heavenly  body 
relative  to  the  sun  are,  if  we  neglect  its  mass  in  comparison  with 
that  of  the  sun, 

dV      *V 

d('  +  r"       °' 


If  we  differentiate  the  first  of  these  equations,  we  get 

dV_3^V    dr^__  k2   M_ 
~ti*~''~7^'  dt       r'3'  dt 

Differentiating  again,  we  find 

JV_/^_12P/d/\2      3^   dV\   ,6^   dS_    cM 
dt  ~~\r'6    "   r'5  \  dt  I   'h  /*  '  ~dP  /*    h  r'*  '  dt  '  dt' 

Writing  y  instead  of  x,  we  shall  have  the  expressions  for  -^   and 

/l^  r  Qv 

-j->  Substituting  these  values  of  the  differential  coefficients  in  equa- 
tions (21),  and  the  corresponding  expressions  for  y  and  y",  and 
putting 


ORBIT   OF   A    HEAVENLY    BODY.  175 

-i__i^_i_^L  ^'+i/l     12/rf/y      3    dy\ 

2>'        24-          +  2'6-3  +^~  •*•' 


T"          r"3  T"*     drr 

'~' 


„  _1__,1__  ,    ,  3      d 

*'3  ~t"2'        +  ^''~5"          +4"  "' 


___         _  . 

~yfc       6^r'3nV*     dt  "    " 

we  obtaiD 


From  these  equations  we  easily  derive 


> 

Civ 

'  '  (23) 


.  . 

The  first  members  of  these  equations  are  double  the  areas  of  the 
triangles  formed  by  the  radii-vectores  and  the  chords  of  the  orbit 
between  the  places  of  the  comet  or  planet.  Thus, 

y'z-s'y  =|W],  sV-a/y^ry],  y"x  -  J'y  =  [n"],  (24) 
and  x'dy'  —  y'dx'  is  double  the  area  described  by  the  radius-vector 
during  the  element  of  time  dt>  and,  consequently,  —  ,  -  ia 

double  the  areal  velocity.  Therefore  we  shall  have,  neglecting  the 
mass  of  the  body, 


iii  which  p  is  the  semi-parameter  of  the  orbit.     The  equations  (23  j, 
therefore,  become 


[r/]  =  bk  V>,  [//']  =  b"k  V~p,  [rr"]  -  (aW 

Substituting  for  a,  6,  a",  b"  their  values  from  (22),  we  find,  since 


176  THEORETICAL   ASTRONOMY. 


(25) 


From  these  equations  the  values  of  n  =  ^  —  J  an(^  n"  ==  r  —  "n  ma7 
be  derived  ;  and  the  results  are 


r"  ?  A;r'*  '  dt  '" 


which  values  are  exact  to  the  third  powers  of  the  time,  inclusive. 

In  the  case  of  the  orbit  of  the  earth,  the  term  of  the  third  order. 

dR' 

being  multiplied  by  the  very  small  quantity  —j-  >  is  reduced  to  a 

dt 

superior  order,  and,  therefore,  it  may  be  neglected,  so  that  in  this 
case  we  shall  have,  to  the  same  degree  of  approximation  as  in  (26), 


n         [r'r"~\ 
From  the  equations  (26)  or  from  (25),  since  —  ,  =  TTTT,  we  find 


i  4 _    i    i 

e       -^T~ 

Since  this  equation  involves  r'  and  -3-,  it  is  evident  that  the  value 

dt 

of  ~,  in  the  case  of  an  orbit  wholly  unknown,  can  be  determined 

Tl 

only  by  successive  approximations.  In  the  first  approximation  to 
the  elements  of  the  orbit  of  a  heavenly  body,  the  intervals  between 
the  observations  will  usually  be  small,  and  the  series  of  terms  of  (28) 
will  converge  rap'dly,  so  that  we  may  take 

n r 

n"  ~  T"' 


ORBIT   OF   A   HEAVENLY   BODY,  177 

and  similarly 

Wf  =  ^' 
Hence  the  equation  (11)  reduces  to 

l>"  =  ~M'p.  (29) 

It  will  be  observed,  further,  that  if  the  intervals  between  the  observa- 
tions are  equal,  the  term  of  the  second  order  in  equation  (28) 

vanishes,  and  the  supposition  that  — ,  =  — ,  is  correct  to  terms  of  the 

71  T 

third  order.  It  will  be  advantageous,  therefore,  to  select  observa- 
tions whose  intervals  approach  nearest  to  equality.  But  if  the 
observations  available  do  not  admit  of  the  selection  of  those  which 
give  nearly  equal  intervals,  and  these  intervals  are  necessarily  very 
unequal,  it  will  be  more  accurate  to  assume 

JL-  -JL 

n"  ~~  N'" 

and  compute  the  values  of  N  and  Nff  by  means  of  equations  (9), 
since,  according  to  (27)  and  (28),  if  r'  does  not  differ  much  from  R', 
the  error  of  this  assumption  will  only  involve  terms  of  the  third 
order,  even  when  the  values  of  r  and  rfr  differ  very  much. 

Whenever  the  values  of  p  and  p"  can  be  found  when  that  of  their 
ratio  is  given,  we  may  at  once  derive  the  corresponding  values  of  r 
and  r",  as  will  be  subsequently  explained. 

The  values  of  r  and  r"  may  also  be  expressed  in  terms  of  r '  by 
means  of  series,  and  we  have 

j  —  —    —  +*  dy    r"2  _  & 
„         ,       dr'     r  dV     ra 

from  which  we  derive 


k       w 
neglecting  terms  of  the  third  order.     Therefore 

dr'_k(r"-r)t 

~dt~    ~T+V'~' 

12 


178  THEORETICAL   ASTEONOMY. 

and  when  the  intervals  are  equal,  this  value  is  exact  to  terms  of  the 
fourth  order.     We  have,  also, 


which  gives 

r)!_r£.  (31) 


Therefore,  when  r  and  r"  have  been  determined  by  a  first  approxi- 
mation, the  approximate  values  of  r'  and  -^-  are  obtained  from  these 
equations,  by  means  of  which  the  value  of  —  may  be  recomputed 
from  equation  (28).  We  also  compute 

N  _#.B"sin(0"-0')  (32) 


N"    '   £K'sm(C)'—  0) 
and  substitute  in  equation  (11)  the  values  of  —  and  -^  thus  found. 


n"         N 
designate  by  M  the  ratio  of  the  cui 

we  have 


If  we  designate  by  M  the  ratio  of  the  curtate  distances  to  and  //', 


NR 


In  the  numerical  application  of  this,  the  approximate  value  of  p  will 
be  used  in  computing  the  last  term  of  the  second  member. 

In  the  case  of  the  determination  of  an  orbit  when  the  approximate 

elements  are  already  known,  the  value  of  -77  may  be  computed  from 

n  __rV'sin(0"  —  iQ  ,„, 

"~~       '' 


and  that  of  -^  from  (32);  and  the  value  of  M  derived  by  means  of 
these  from  (33)  will  not  require  any  further  correction. 

65.  When  the  apparent  path  of  the  body  is  such  that  the  value 
of  Mf,  as  derived  from  the  first  of  equations  (10),  is  either  indeter- 
minate or  greatly  aifected  by  errors  of  observation,  the  equations  (15) 
and  (16)  must  be  employed.  The  last  terms  of  these  equations  may 
be  changed  to  a  form  which  is  more  convenient  in  the  approximations 
to  the  value  of  the  ratio  of  p"  to  p. 

Let   Yt  Yf,  Y"  be  the  ordinates  of  the  sun  when  the  axis  of 


ORBIT  OF  A  HEAVENLY  BODY.  179 

abscissas  is  directed  to  that  point  in  the  ecliptic  whose  longitude  is 
^',  and  we  have 

Y  =E  sin(O   —  *'), 

Y'  =E'  sin  (0'—  A'), 


Now,  in  the  last  term  of  equation  (15),  it  will  be  sufficient  to  put 

n_        N 

n"~  Ne" 

and,  introducing  Y,  Y',  Y",  it  becomes 


cosec      - 


It  now  remains  to  find  the  value  of  -77-     From  the  second  of  equa- 
tions (26)  we  find,  to  terms  of  the  second  order  inclusive, 


We  have,  also, 
and  hence 


Therefore,  the  expression  (35)  becomes 


But,  according  to  equations  (5), 

NY—Y'-\-N" 
and  the  foregoing  expression  reduces  to 


sin  (/'-/)    ' 
since  Y'  =  Rf  sin  (Or  —  Af).     Hence  the  equation  (15)  becomes 

1  \^sin(/-0Q    (    . 

-*  •  (36) 


180  THEORETICAL   ASTRONOMY. 

If  we  put 

...  _  _n_    sin  (A'  —  ^) 
°~~"ri"  *sin(/l"  —  /)' 

JF-!  _  .  *"  .  ^  ' 

* 


sn      - 
we  have 

£  =  M=M9F.  (37) 

Let  us  now  consider  the  equation  (16),  and  let  us  designate  by  X, 
X'y  X"  the  abscissas  of  the  earth,  the  axis  of  abscissas  being  directed 
to  that  point  of  the  ecliptic  for  which  the  longitude  is  Or,  then 

X  =  JR  cos  (0  —  00, 
X'=Kf, 

X"=JR"eos(0"—  00. 

It  will  be  sufficient,  in  the  last  term  of  (16),  to  put 

Jl      JL 
n"  ~  N"  ' 

and  for  —77-  its  value  in  terms  of  N"  as  already  found.     Then,  since 


this  term  reduces  to 
-l^rfV+yW1        -1\  _  ^ta 

B  rn  ^  *  J\  pi        E,3]  tan  p  cog  y,_  0,  j  _ 

and  if  we  put 

__n_     tan  /3'  cos  (A  —  Qr)  —  tan  /?  cos  (A;  —  Q;) 
0  ~~  "n77  '  tan  ft"  cos  (A'  —  Q')  —  tan  pr  cos  (A"  —  Q')'  (38) 


tan^'cos  (X—  0')—  tan/3oos(X'—  Q') 
the  equation  (16)  becomes 

^F'.  (39) 


In  the  numerical  application  of  these  formulae,  if  the  elements  are 
not  approximately  known,  we  first  assume 


n 


when  the  intervals  are  nearly  equal,  and 


OKBIT   OF  A   HEAVENLY   BODY.  181 


n"  "  N"  ' 

as  given  by  (32),  when  the  intervals  are  very  unequal,  and  neglect 
the  factors  F  and  F'.  The  values  of  to  and  pfr  which  are  thus  ob- 
tained, enable  us  to  find  an  approximate  value  of  rf,  and  with  this  a 

more  exact  value  of  —  ,7  may  be  found,  and  also  the  value  of  F  or  Ff. 

Whenever  equation  (11)  is  not  materially  affected  by  errors  of 
observation,  it  will  furnish  the  value  of  M  with  more  accuracy  than 
the  equations  (37)  and  (39),  since  the  neglected  terms  will  not  be  so 
great  as  in  the  case  of  these  equations.  In  general,  therefore,  it  is  to 
be  preferred,  and,  in  the  case  in  which  it  fails,  the  very  circumstance 
that  the  geocentric  path  of  the  body  is  nearly  in  a  great  circle,  makes 
the  values  of  F  and  F'  differ  but  little  from  unity,  since,  in  order 
that  the  apparent  path  of  the  body  may  be  nearly  in  a  great  circle, 
r'  must  differ  very  little  from  Rf. 

66.  When  the  value  of  M  has  been  found,  we  may  proceed  to 
determine,  by  means  of  other  relations  between  p  and  ptr9  the  values 
of  the  quantities  themselves. 

The  co-ordinates  of  the  first  place  of  the  earth  referred  to  the  third, 

are 

x,  —  R"  cos  Q  "  —  R  cos  Q  ,  ,  .^ 

y,  =  E"smO"  —  KsmQ. 

If  we  represent  by  g  the  chord  of  the  earth's  orbit  between  the  places 
corresponding  to  the  first  and  third  observations,  and  by  G  the  longi- 
tude of  the  first  place  of  the  earth  as  seen  from  the  third,  we  shall 
have 

x,  =  g  cos  G,  y,  =  ff  sin  G, 

and,  consequently, 

12"cos(0"—  0)  —  jR  =  ?cos(£—  O), 
jR"sin(©"—  0)  =gsm(G—®'). 

If  ^  represents  the  angle  at  the  earth  between  the  sun  and  comet 
at  the  first  observation,  and  if  we  designate  by  w  the  inclination  to 
the  ecliptic  of  a  plane  passing  through  the  places  of  the  earth,  sun, 
and  comet  or  planet  for  the  first  observation,  the  longitude  of  the 
ascending  node  of  this  plane  on  the  ecliptic  will  be  O,  and  we  shall 
have,  in  accordance  with  equations  (81)^ 

cos  *  =  cos  fi  cos  (A  —  0), 
sin  4  cos  w  =  cos  /?  sin  (A  —  0), 
sin  4  sin  w  —  sin  & 


182  THEORETICAL   ASTRONOMY. 

from  which 

tan/5 
tan  w  =  -T 


sin  (A  —  Q)' 

tan(A-G) 

tan  4  = . 

costy 

Since  cos  ft  is  always  positive,  cos  ^  and  cos  (A  —  0)  must  have  the 
same  sign;  and,  further,  a//  cannot  exceed  180°. 

In  the  same  manner,  if  w"  and  ij/'  represent  analogous  quantities 
for  the  time  of  the  third  observation,  we  obtain 

tan/5" 
tanw"  =  - 


sin  (A"  —  Q")' 

^  (43) 


cos  4"  =  cos  f  cos  (A"  —  0"). 
We  also  have 


which  may  be  transformed  into 

ra  =  G>  sec£  —  R  cos^  +  .ft2  sin**;  (44) 

and  in  a  similar  manner  we  find 

r"2  =  (p"  sec  /3"  —  #'  cos  V')1  +  #"  sin2  *".  (45) 

Let  K  designate  the  chord  of  the  orbit  of  the  body  between  the  first 
and  third  places,  and  we  have 

X>  =   (*"  _  ,.).  _|_    (f  _  y).  +    (2"  _  f  )«. 

But 

»  =  p  cos  A  —  R  cos  0, 
2/=i0smA  —  E  sin  O, 
z  =  p  tan  /5, 

and,  since  />"  =  Mp, 


f  =  Mp  sin  X"  —  jR"  sin  O", 


from  which  we  derive,  introducing  g  and  (r, 


a/'  —  #  =  JW/>  cos  A"  —  />  cos  A  —  g  cos  $, 
y"  —  V  =  Mp  sm  *-"  —  jO  sin  A  —  g  sin  G, 
z"  —  z  =  Mp  tan/5"—  p  tan  /?. 


Let  us  now  put 


ORBIT   OF  A   HEAVENLY   BODY.  183 

Mp  cos  A"  —  p  cos  A  =  ph  cos  C  cos  H, 
Mp  sin  A"  —  />  sin  A  =  ph  cos  C  sin  .5",  (46) 

Mp  tan  /3"  —  p  tan  fi=  ph  sin  C. 
Then  we  have 

#"  —  x  —  /oA  cos  C  cos  IT  —  #  cos  6r, 

2/"  —  y  =  ph  cos  C  sin  H  —  g  sin  (2, 

z"  —  z  =  ph  sin  C. 

Squaring  these  values,  and  adding,  we  get,  by  reduction, 

x2  =  />2/i2  —  2g  ph  cos  C  cos  (  G  —  .H")  +  f  ;  (47) 

and  if  we  put 

cos  C  cos  (  G  —  H}  =  cos  ?>,  (48^ 

we  have 

x2  =  (/>A  —  y  cos  ^)2  -f-  <72  sin2  y.  (49) 

If  we  multiply  the  first  of  equations  (46)  by  cos^",  and  the 
second  by  sin  A",  and  add  the  products;  then  multiply  the  first  by 
sin  X",  and  the  second  by  cos  Kr,  and  subtract,  we  obtain 

h  cos  C  cos  (H—  A")  =  M  —  cos  (A"  —  A), 

h  cos  C  sin  (H  -  n  =  sin  (/"  —  A),  (50) 

/i  sin  C  =M  tan  /3"  —  tan  ft 

by  means  of  which  we  may  determine  A,  f  ,  and  H. 
Let  us  now  put 

g  sin  ^  =  Ay 
R  sin  4*  =  B,  h  cos  £  =  b, 


R"  sin  4"  =  JB",  -  6",  (51) 

#  cos  ?>  —  6jR  cos  ^  =  c,  g  cos?  —  b"R"  cos  4"  =  c", 

jO/i  —  gr  cos  p  =  d, 

and  the  equations  (44),  (45),  and  (49)  become 


The  equations  thus  derived  are  independent  of  the  form  of  the 
orbit,  and  are  applicable  to  the  case  of  any  heavenly  body  revolving 
around  the  sun.  They  will  serve  to  determine  r  and  r"  in  all  cases 
in  which  the  unknown  quantity  d  can  be  determined.  If  p  is  known, 


184  THEORETICAL   ASTRONOMY. 

d  becomes  known  directly;  but  in  the  case  of  an  unknown  orbit, 
these  equations  are  applicable  only  when  p  or  d  may  be  determined 
directly  or  indirectly  from  the  data  furnished  by  observation. 

67.  Since  the  equations  (52)  involve  two  radii-vectores  r  and  r" 
and  the  chord  x  joining  their  extremities,  it  is  evident  that  an  addi- 
tional equation  involving  these  and  known  quantities  will  enable  us 
to  derive  d,  if  not  directly,  at  least  by  successive  approximations. 
There  is,  indeed,  a  remarkable  relation  existing  between  two  radii- 
vectores,  the  chord  joining  their  extremities,  and  the  time  of  describing 
the  part  of  the  orbit  included  by  these  radii-vectores.  In  general, 
the  equation  which  expresses  this  relation  involves  also  the  semi- 
transverse  axis  of  the  orbit;  and  hence,  in  the  case  of  an  unknown 
orbit,  it  will  not  be  sufficient,  in  connection  with  the  equations  (52), 
for  the  determination  of  c?,  unless  some  assumption  is  made  in  regard 
to  the  value  of  the  semi-transverse  axis.  For  the  special  case  of 
parabolic  motion,  the  semi-transverse  axis  is  infinite,  and  the  result- 
ing equation  involves  only  the  time,  the  two  radii-vectores,  and  the 
chord  of  the  part  of  the  orbit  included  by  these.  It  is,  therefore, 
adapted  to  the  determination  of  the  elements  when  the  orbit  is  sup- 
posed to  be  a  parabola,  and,  though  it  is  transcendental  in  form,  it 
may  be  easily  solved  by  trial.  To  determine  this  expression,  let  us 
resume  the  equations 

lc(t-T} 


1/2  jt 
and,  for  the  time  t"} 


=  tan  %v  -j-  I  tan*  ^v 


1/2  gi 

Subtracting  the  former  from  the  latter,  and  reducing,  we  obtain 

1/2  gf      ~~  cos  J-y"  cos  ^v  \  ~q       cos  ^v"  cos  £i>      q  / 
and,  since  r  =  q  sec2^,  this  gives 

On/ 1  t  '  C )  Sill  77  \  t/  i/  /    I/    **       I  r  /    i  •*    s    tt  ~\     /       /7  I       /"  r*  r»\ 

— ^— = — -  = i:^ 7=r^ 1  r  -|-  r'+  cos  I  (v — vjvrr     •  (53) 

1/2  1/g  \  / 

But  we  have,  also,  from  the  triangle  formed  by  the  chord  vc  and  the 
radii-vectores  r  and  rr/, 

x2  =  r2  -f-  r"2  —  2rr"  cos  (?/' —  v) 
=  (r  +  r")2  —  4rr"  cos'  J  W  —  v). 


PARABOLIC  ORBIT.  185 

Therefore, 


21/W" 
Let  us  now  put 

r  +  r"  +  x  =  m3,  r  +  r"—  x  =  n», 

m  and  n  being  positive  quantities.     Then  we  shall  have 

r  +  r"=J(m'+*«), 
2  cos  J  (v"  —  v)  Vrr"  =  ±  mn  ; 

and,  since  m  and  n  are  always  positive,  it  follows  that  the  upper  sign 
must  be  used  when  v"  —  v  is  less  than  180°,  and  the  lower  ^ign  when 
v"  —  v  is  greater  than  180°.  Combining  the  last  equation  with  (53), 
the  result  is 


3k  (f  -  0  =  (m«  +  »•  ±  m»).  (55) 

y  2g 

Now  we  have 

sin  |-  (v"  —  v)  =  sin  £v"  cos  ^v  —  cos  ^v"  sin  ^v. 
Squaring  this,  and  reducing,  we  get 

sin2  \  (v"  —  v)  =  cos2  ^v  -\-  cos2  £v"  —  2  cos  Jv"  cos  |v  cos  J  (t^     •  v), 

or,  introducing  r  and  q, 

•  9  1  s  n        \       Q   , 
Bin1  i  (w  "  —  v)  =1  -f 

Therefore, 

i    "  —          --7= 

introducing  this  value  into  equation  (55),  we  find 


sin  i  (v"  —  v)  =  --=%  <>  =1=  n). 


Replacing  m  and  n  by  their  values  expressed  in  terms  of  r,  r",  and 
x,  this  becomes 

6k  (if'  -?>  =  (r  +  r"  +  x)t  T  (r  +  r"  -  x)t,  (56) 

the  upper  sign  being  used  when  v"  —  v  is  less  than  180°.  This 
equation  expresses  the  relation  between  the  time  of  describing  any 
parabolic  arc  and  the  rectilinear  distances  of  its  extremities  from  each 
other  and  from  the  sun,  and  enables  us  at  once,  when  three  of  these 
quantities  are  given,  to  find  the  fourth,  independent  of  either  the 


186  THEORETICAL   ASTRONOMY. 

perihelion  distance  or  the  position  of  the  perihelion  with  respect  to 
the  arc  described. 

68.  The  transcendental  form  of  the  equation  (56)  indicates  that, 
when  either  of  the  quantities  in  the  second  member  is  to  be  found, 
it  must  be  solved  by  successive  trials ;  and,  to  facilitate  these  approxi- 
mations, it  may  be  transformed  as  follows : — 

Since  the  chord  x  can  never  exceed  r  +  r",  we  may  put 

7Ippr  =  sin/,  (57) 

and,  since  x,  r,  and  r"  are  positive,  sin  fr  must  aiways  be  positive. 
The  value  of  f  must,  therefore,  be  within  the  limits  0°  and  180°. 
From  the  last  equation  we  obtain 

cosV=>  +  r")2-< 

and  substituting  for  Jt2  its  value  given  by 

x2  =  (r  +  r")2  —  4rr"  cos2  £  (v"  —  v), 
this  becomes  * 

4rr"  cos2  W  —  v) 


(r  +  r,,y 

Therefore,  we  have 

cos  Y'  =  cos  |  (y"  —  v)  —       .,,  (58) 

and  also 

"r/"" 


Hence  it  appears  that  when  v"  —  v  is  less  than  180°,  f  belongs  to 
the  first  quadrant,  and  that  when  v"  —  v  is  greater  than  180°,  cosf 
is  negative,  and  f'  belongs  to  the  second  quadrant. 

Tf  we  introduce  f'  into  the  expressions  for  m2  and  n2,  they  becoma 


^  =  (r  +    ')(!-  sin      , 
which  give 

m*=  (r  +  r")  (cos  tf  +  sin  J/)«, 
W2  =  (r  4.  /')  (-4-  cos  J/  ip  sin  yj  ; 

and,  since  f  ig  greater   than  90°  when  v"  —  v  exceeds  180°,  the 
equation  (56)  becomes 

=  (cosy  +  sin  £/)•-  (cos  J/-siny)« 


(r 


PARABOLIC   ORBIT.  187 

From  this  equation  we  get 

—  =  6  cos2  $  sin  %/  -f-  2  sin*  $/, 

or 

/>    f 

—  ^TTT-  =  6  sin !/—  4  sin8  tf ; 

and  this,  again,  may  be  transformed  into 
6r' 


(60) 

Let  us  now  put 


sn 

" 


or 

sin  \*f'  =  l/  2  sin  #, 
and  we  have 


=  3  sin  #  —  4  sin3  x  =  sin  3#.  (62) 


is  less  than  180°,  f  must  be  less  than  90°,  and 
hence,  in  this  case,  sin  x  cannot  exceed  the  value  |,  or  x  must  be 
within  the  limits  0°  and  30°.  When  v"  —  v  is  greater  than  180°, 
the  angle  f  is  within  the  limits  90°  and  180°,  and  corresponding  to 
these  limits,  the  values  of  sin  x  are,  respectively,  \  and  \  \/%.  Hence, 
in  the  case  that  vrr  —  v  exceeds  180°,  it  follows  that  x  must  be  within 
the  limits  30°  and  45°. 
The  equation 


1/20 

is  satisfied  by  the  values  3#  and  180°  —  3x;  but  when  the  first  gives 
x  less  than  15°,  there  can  be  but  one  solution,  the  value  180° — 3x 
being  in  this  case  excluded  by  the  condition  that  3x  cannot  exceed 
135°.  When  x  is  greater  than  15°,  the  required  condition  will  be 
satisfied  by  3x  or  by  180°  —  3#,  and  there  will  be  two  solutions, 
corresponding  respectively  to  the  cases  in  which  vff  —  v  is  less  than 
180°,  and  in  which  v"  —  v  is  greater  than  180°.  Consequently, 
wnen  it  is  not  known  whether  the  heliocentric  motion  during  the 
interval  trf  —  t  is  greater  or  less  than  180°,  and  we  find  3x  greater 
than  45°,  the  same  data  will  be  satisfied  by  these  two  different 
solutions.  In  practice,  however,  it  is  readily  known  which  of  the 


188  THEOEETICAL   ASTRONOMY. 

two  solutions  must  be  adopted,  since,  when  the  interval  t"  —  t  is  not 
very  large,  the  heliocentric  motion  cannot  exceed  180°,  unless  the 
perihelion  distance  is  very  small;  and  the  known  circumstances  will 
generally  show  whether  such  an  assumption  is  admissible. 

We  shall  now  put 

o/ 

•n  =  —  —  -,  (63) 


sin  3*=  -L  (64) 

T/8 


and  we  obtain 

We  have,  also, 

sin  ^/  =  |/2  sin  x, 

and  hence 

cos  \r'  =  i/l  —  2  sin2  x  =  I/cos  2x. 
Therefore 

sin     =  2t  sin  x  V  cos 


and,  since  JC  =  (r  -|-  r")  sin  f,  we  have 

x  =  2%  (r  +  r")  sin  a;  I/cos  2x. 
If  we  put 

3sin#    /  -  ^-  ,«.x 

^  =    .    o    V  cos  2x,  (b5) 

sinoa; 

the  preceding  equation  reduces  to 


From  equation  (64)  it  appears  that  rj  must  be  within  the  limits  0 
and  Ji/8-  We  may,  therefore,  construct  a  table  which,  with  ^  as 
the  argument,  will  give  the  corresponding  value  of  //,  since,  with  a 
given  value  of  37,  3x  may  be  derived  from  equation  (64),  and  then 
the  value  of  //  from  (65).  Table  XI.  gives  the  values  of  //  corre- 
sponding to  values  of  r]  from  0.0  to  0.9. 

69.  In  determining  an  orbit  wholly  unknown,  it  will  be  necessary 
to  make  some  assumption  in  regard  to  the  approximate  distance  of 
the  comet  from  the  sun.  In  this  case  the  interval  t"  —  t  will  gene- 
rally be  small,  and,  consequently,  x  will  be  small  compared  with  r 
and  r".  As  a  first  assumption  we  may  take  r  =  1,  or  r  4  r"  =  2, 
and  fj.  =  1,  and  then  find  x  from  the  formula 


FiRABOLIC  ORBIT.  189 

With  this  value  of  x  we  compute  d,  r,  and  r"  by  means  of  the 
equations  (52).  Having  thus  found  approximate  values  of  r  and  r", 
we  compute  y  by  means  of  (63),  and  with  this  value  we  enter  Table 
XI.  and  take  out  the  corresponding  value  of  p.  A  second  value 
for  K  is  then  found  from  (66),  with  which  we  recompute  r  and  r"  ',  and 
proceed  as  before,  until  the  values  of  these  quantities  remain  un- 
changed. The  final  values  will  exactly  satisfy  the  equation  (56), 
and  will  enable  us  to  complete  the  determination  of  the  orbit. 

After  three  trials  the  value  of  r  -f  r"  may  be  found  very  nearly 
correct  from  the  numbers  already  derived.  Thus,  let  y  be  the  true 
value  of  log  (r  -j-  r"),  and  let  A?/  be  the  difference  between  any 
assumed  or  approximate  value  of  y  and  the  true  value,  or 

2/0  =  y  +  Ay. 

Then  if  we  denote  by  y0f  the  value  which  results  by  direct  calculation 
from  the  assumed  value  yQ,  we  shall  have 


2A>'  —  y0  =/(y0)  = 

.Expanding  this  function,  we  have 

2/o'  —  2/o  =/(y)  +  4  *y  -4-  £  Ay*  +  Ac. 


But,  since  the  equations  (52)  and  (66)  will  be  exactly  satisfied  when 
the  true  value  of  y  is  used,  it  follows  that 


and  hence,  when  AT/  is  very  small,  so  that  we  may  neglect  terms  ot 
the  second  order,  we  shall  have 

Vo'  —  y,  =  4  Ay  =  4  (y0  —  y*). 

Let  us  now  denote  three  successive  approximate  values  of  log  (r  +  r") 
b7  2A»  2A/>  2/o">  and  let 

y0'  —  yc  =  a>  y"  —  &'  =  «'•• 

then  we  shall  have 

a  =  A  (y0  —  y), 
a'  =  -4  (&'-?). 

Eliminating  A  from  these  equations,  we  get 

y  (af  —  o)  =  a'y0  -  -  ay0', 

from  which 

„„'  /»'» 

(67; 


190  THEORETICAL   ASTRONOMY. 

Unless  the  assumed  values  are  considerably  in  error,  the  value  of 
y  or  of  log  (r  +  r")  thus  found  will  be  sufficiently  exact ;  but  should 
it  be  still  in  error,  we  may,  from  the  three  values  which  approximate 
nearest  to  the  truth,  derive  y  with  still  greater  accuracy.  In  the 
numerical  application  of  this  equation,  a  and  a'  may  be  expressed  in 
units  of  the  last  decimal  place  of  the  logarithms  employed. 

The  solution  of  equation  (56),  to  find  t"  —  t  when  JC  is  known,  is 
readily  effected  by  means  of  Table  VIII.  Thus  we  have 

=  sin  3#. 


l/2(r-f-*-")f 
and,  when  f1  is  less  than  90°,  if  we  put 


.     ----       .         /~i 

sin  •/ 
we  get 

T'  =  -i  1/2  N  sin  r'  (r  +  r")  t,  (68) 

or 


When  f  exceeds  90°,  we  put 

N'  =  sin  3s, 
and  we  have 

/  =  4  1/2  JF(r  +  /')*» 

in  which  log  $  j/2  =  9.6733937.  With  the  argument  f  we  take 
from  Table  VIII.  the  corresponding  value  of  N  or  JV7,  and  by 
means  of  these  equations  r'  =  k  (t"  —  t)  is  at  once  derived. 

The  inverse  problem,  in  which  r'  is  known  and  K  is  required,  may 
also  be  solved  by  means  of  the  same  table.  Thus,  we  may  for  a  first 
approximation  put 

*  =  T'  1/2. 

and  with  this  value  of  K  compute  d,  r,  and  r"  .  The  value  of  f1  is 
then  found  from 


and  the  table  gives  the  corresponding  value  of  N  QT  JV7.     A  second 
approximation  to  x  will  be  given  by  the  equation 


S_ 

1/2 '  NVT- 


PAEABOLIC  CEBIT.  101 

or  by 

3          /sin/ 

'v^'^'i/T+y' 

in  which  log  — =  =  0.3266063.     Then  we  recompute  d,  r,  and  r", 

v  2 

and  proceed  as  before  until  x  remains  unchanged.     The  approxima- 
tions are  facilitated  by  means  of  equation  (67). 
It  will  be  observed  that  d  is  computed  from 


and  it  should  be  known  whether  the  positive  or  negative  sign  must 
be  used.  It  is  evident  from  the  equation 

d  =  ph  —  g  cos  y>, 

since  /?,  h,  and  g  are  positive  quantities,  that  so  long  as  <p  (which 
must  be  within  the  limits  0°  and  180°)  exceeds  90°,  the  value  of  d 
must  be  positive ;  and  therefore  <p  must  be  less  than  90°,  and  g  cos  <p 
greater  than  ph,  in  order  that  d  may  be  negative.  The  equation  (47) 
shows  that  when  x  is  greater  than  g,  we  have 

g  cos  <p  <  %ph, 

and  hence  d  must  in  this  case  be  positive.  But  when  x  is  less  than 
g,  either  the  positive  or  the  negative  value  of  d  will  answer  to  the 
given  value  of  ^>,  and  the  sign  to  be  adopted  must  be  determined 
from  the  physical  conditions  of  the  problem. 

If  we  suppose  the  chords  g  and  x  to  be  proportional  to  the  linear 
velocities  of  the  earth  and  comet  at  the  middle  observation,  we  have, 
the  eccentricity  of  the  earth's  orbit  being  neglected, 


=  SrV 


which  shows  that  H  is  greater  than  g}  and  that  d  is  positive,  so  long 
as  rf  is  less  than  2.  The  comets  are  rarely  visible  at  a  distance  from 
the  earth  which  much  exceeds  the  distance  of  the  earth  from  the  sun, 
and  a  comet  whose  radius-vector  is  2  must  be  nearly  in  opposition  in 
order  to  satisfy  this  condition  of  visibility.  Hence  cases  will  rarely 
occur  in  which  d  can  be  negative,  and  for  those  which  do  occur  it 
will  generally  be  easy  to  determine  which  sign  is  to  be  used.  How- 
ever, if  d  is  very  small,  it  may  be  impossible  to  decide  which  of  the 
two  solutions  is  correct  without  comparing  the  resulting  elements 
with  other  and  more  distant  observations. 


192  THEOKETICAL   ASTKONOMY. 

70.  When  the  values  of  r  and  r"  have  been  finally  determined,  as 
just  explained,  the  exact  value  of  d  may  be  computed,  and  then  we 
have 

_  d  +  g  cos  <p 

~h~  (70) 

P"  =  MP) 

from  which  to  find  p  and  p". 

According  to  the  equations  (90)1?  we  have 

r  cos  b  cos  (I  —  O)  =  p  cos  (A  —  Q)  —  R, 
r  cos  b  sin  (I  —  O)  =  p  sin  (A  —  Q),  (71) 

r  sin  6  =  />  tan  /?, 

and  also 

r"  cos  b"  cos  (r  —  O")  =  p"  cos  (A"  —  O")  —  Rr, 

r"  cos  b"  sin  (r  —  O")  =  p"  sin  (A"  —  0"),  (72) 

r"sin&"  =ytan/9", 

in  which  I  and  Z;/  are  the  heliocentric  longitudes  and  6,  b"  the  corre- 
sponding heliocentric  latitudes  of  the  comet.  From  these  equations 
we  find  r,  r",  I,  lff,  6,  and  b"  •  and  the  values  of  r  and  r"  thus  found, 
should  agree  with  the  final  values  already  obtained.  When  I"  is  less 
than  I,  the  motion  of  the  comet  is  retrograde,  or,  rather,  when  the 
motion  is  such  that  the  heliocentric  longitude  is  diminishing  instead 
of  increasing. 

From  the  equations  (82)w  we  have 

±  tan  i  sin  (I  —  £ )  =  tan  6, 
±  tan  i  sin  (I"—  &  )  =  tan  b", 

which  may  be  written 

zt  tan  i(sin  (I  —  x)  cos  (x  —  &)  -f  sin  (re  —  &)  cos  (I  —  x))  =  tan  b, 
±  tan  i  (sin  (£" —  a?)  cos  (a;  —  &  )  -f  sin  (re  —  £  )  cos  (r —  #))  =  tan  b". 

Multiplying  the  first  of  these  equations  by  sin  (I"  —  x)9  and  the  second 
by  —  sin  (I  —  a?),  and  adding  the  products,  we  get 

it  tan  i  sin  (x  —  Q, )  sin  (ff  -  -  F)  =  tan  b  sin  (I"  —  x)  —  tan  b"  sin  (I  —  a:) ; 

and  in  a  similar  manner  we  find 

it  tan  i  cos  (a;  —  &  )  sin  (I"  —  I)  =  tan  6"  cos  (I  —  x)  —  tan  b  cos  (I"  —  x). 

Now,  since  x  is  entirely  arbitrary,  we  may  put  it  equal  to  I,  and  we 
have 


PAKABOLIC   CEBIT.  193 

tan  i  sin  (I  —  ft  )  =  db  tan  b, 

.       n       •'  tan  6"—  tan  6  cos  (?'  —  I)  (74) 


the  lower  sign  being  used  when  it  is  desired  to  introduce  the  distinc- 
tion of  retrograde  motion. 

The  formulae  will  be  better  adapted  to  logarithmic  calculation  if 
we  put  x  =  \(l"+  0,  whence  l"—x=l(l"  —  l)  and  I—  x=\(l—  I")-, 
and  we  obtain 

taut  sinQ(r-f  Q  —  ft)  =  ±:  5  -   sin  (^+6)    - 

2  cos  6  cos  6"  cos  J  (/"  —  I) 
'  '    (  jy      £\ 

tanicos(Kr  +  0  -  O)  =  ±  g  CQ5i  SC'0P8V,  ~  |^. 

These  equations  may  also  be  derived  directly  from  (73)  by  addition 
and  subtraction.     Thus  we  have 


db  tan  i  (sin  (I"  —  ft  )  -f  sin  (I  —  ft  ))  =  tan  b"  -f  tan  b, 
±  tani(sin(r—  ft)  —  sin (7—  ft))  =  tan&"  —  tan6; 


and,  since 


sin  (r—  ft)  -f  sin  (I  —  ft)  =  2  sin  £  (f'+  ^  -  2ft  )  cos  i  (r~  O, 
sin(r—  ft)—  sin^—  ft)  =2cosi(^'+  ^  —  2ft)  sin  J(^'—  0» 


these  become 

<d.(,(r+l)-a)-±"_. 


which  may  be  readily  transformed  into  (75).  However,  since  b  and 
b"  will  be  found  by  means  of  their  tangents  in  the  numerical  appli- 
cation of  equations  (71)  and  (72),  if  addition  and  subtraction  loga- 
rithms are  used,  the  equations  last  derived  will  be  more  convenient 
than  in  the  form  (75). 

As  soon  as  ft  and  i  have  been  computed  from  the  preceding  equa- 
tions, we  have,  for  the  determination  of  the  arguments  of  the  latitude 
u  and  u", 


COS  I  COS  I 

Now  we  have 

u  =  v  -f-  o>, 

in  which  CO  =  TC  —  ft  in  the  case  of  direct  motion,  and  CD  =•  ft  —  K 

13 


194  THEORETICAL   ASTRONOMY. 

when  the  distinction  of  retrograde  motion  is  adopted;  and  we  shall 
have 

if.-ea-V'—  4 
and,  consequently, 

x2  =  r2  +  r"2  -     2rr"  cos  (u"  —  u),  (78) 

or 

x2  =  (/'  —  r  cos  O"  —  it))2  +  r8  sin2  («"  —  u).  (79) 

The  value  of  K  derived  from  this  equation  should  agree  with  that 
already  found  from  (66). 
We  have,  further, 

r  =  q  sec2  £(u  —  «),  **'  =  <?  sec2  3  (X'  —  "Of 

or 

-  7=  COS  ?(U  -  0>)  ==:  -  T=,  —  -=  COS  i  (w"  —  «>)  = 

T/g  1/r  1/5 

By  addition  and  subtraction,  we  get,  from  these  equations, 

—  «)  +  cos  l(w  —  o>))  =  —  =  4- 


frum  which  we  easily  derive 

~  cos  HK«"  +«)-»)  cos  K«"-  «)  =  4= 
1/3  t/r 


But 

1  1  1          /     4/7^  _       4 

T7+^~^\X  r 
and  if  we  put 


4f7r 
since  -yf  —  will  not  differ  much  from  1,  ^'  will  be  a  small  angle;  and 

we  shall  have,  since  tan  (45°  +  0')  —  cot  (45°  +  6')  =  2  tan  20', 


PARABOLIC   ORBIT.  195 

Therefore,  the  equations  (80)  become 


1      .    1  e  ,  f  „  .  .  tan  2^ 

—  sin  ' 


sin  |  (ti"  —  u)  Vrr1'  ' 


cos 


cos     w  - 


from  which  the  values  of  q  and  w  may  be  found.     Then  we  shall 
have,  for  the  longitude  of  the  perihelion 


when  the  motion  is  direct,  and 

*  =  &  —  «*, 

when  i  unrestricted  exceeds  90°  and  the  distinction  of  retrograde 
motion  is  adopted. 

It  remains  now  to  find  Ty  the  time  of  perihelion  passage.    We  have 

V  =  U  -  0>,  1/'=u"  -  to. 

With  the  resulting  values  of  v  and  v"  we  may  find,  by  means  of 
Table  VI.,  the  corresponding  values  of  M  (which  must  be  distin- 
guished from  the  symbol  M  already  used  to  denote  the  ratio  of  the 
curtate  distances),  and  if  these  values  are  designated  by  M  and  M  ", 
we  shall  have 

* 


t-T= 

m  m 


or 


m  m 

£f 

in  which  m  =  -f  ,  and  log  CQ  —  9.9601277.     When  v  is  negative,  the 
gi 

corresponding  value  of  M  is  negative.  The  agreement  between  the 
two  values  of  T  will  be  a  final  proof  of  the  accuracy  of  the  numerical 
calculation. 

The  value  of  T  when  the  true  anomaly  is  small,  is  most  readily 
and  accurately  found  by  means  of  Table  VIII.,  from  which  we 
derive  the  two  values  of  ^V  and  compute  the  corresponding  values 
of  T  from  the  equation 


2 

in  which  logg,  =  1.5883273.     When  v  is  greater  than  90°,  we  de- 


196  THEORETICAL   ASTRONOMY. 

rive  the  values  of  N'  from  the  table,  and  compute  the  corresponding 
values  of  T  from 


71.  The  elements  q  and  Tmay  be  derived  directly  from  the  values 
of  r,  r",  and  x,  as  derived  from  the  equations  (52),  without  first 
finding  the  position  of  the  plane  of  the  orbit  and  the  position  of  the 
orbit  in  its  own  plane.  Thus,  the  equations  (80),  replacing  u  and  un 
by  their  values  v  -\-  co  and  v  +  &>",  become 

4»dni  (^  +  tO  sinj  (v"  -  v)  =  ±—* 
V,  Vr       Vr 


Adding  together  the  squares  of  these,  and  reducing,  we  get 


or 

Q  _ " '^~'_~~ " " 

Combining  this  equation  with  (59),  the  result  is 

"       r  -f-  r"  —  > 
and  hence,  since  x  =  (r  +  r")  sin  7-', 

5  =  —  sin2  £  (i/'  —  tO  co 
We  have,  further,  from  (78), 

from  which,  putting 

sinv  =  r^H^t  (84) 

x 

we  derive 

2i  v  rr     •»/•//        >  foK."\ 

cos  v  = sin  J  (v  —  v).  (85; 

x 

Therefore,  the  equation  (83)  becomes 


PARABOLIC  ORBIT.  197 

0  =  J  (r  +  r")  cos2  tf  cos'v,  (86) 

by  means  of  which  q  is  derived  directly  from  r,  r",  and  K,  the  value 
of  v  being  found  by  means  of  the  formula  (84),  so  that  cosv  ia 
positive. 

When  f  cannot  be  found  with  sufficient  accuracy  from  the  equa- 
tion 


we  may  use  another  form.     Thus,  we  have 


which  give,  by  division, 


tan(45°-Hr')  =  \/r_   llllli  (87) 

~r~  §       -''---  X 


In  a  similar  manner,  we  derive 


tan  (45°  +  £")  =  \*_  S/Z  (88> 

In  order  to  find  the  time  of  perihelion  passage,  it  is  necessary  first 
to  derive  the  values  of  v  and  v".  The  equations  (59)  and  (85)  give, 
by  multiplication, 

tan  £  (v"  —  v)  =  tan  -/  cos  v,  (89) 

from  which  v"  —  v  may  be  computed.     From  (82)  we  get 


If  we  put 

tan /'  =  */_-.,  (9u) 

this  equation  reduces  to 

tan  $  (y"  +  v)  =  tan  (/  —  45°)  cot  j  (v"  —  v),  (91) 

and  the  equations  (81)  give,  also, 

tan  i  (t/'  +  t>)  =  cot  i  (i/'  —  v)  sin  2^, 
either  of  which  may  be  used  to  find  v"  -f-  v. 


198  THEORETICAL   ASTRONOMY. 

From  the  equations 

cos  |v  _     1  cos  \v"         1 

Vq         Vr  Vq          Vr" 

by  multiplying  the  first  by  sin  \v"  and  the  second  by  —  sin  \v,  add- 
ing the  products  and  reducing,  we  easily  find 

sin  4  (V'  —  v)  sin  |t> cos  \  (v"  —  v)  _      1 

Vq  Vr 

Hence  we  have 


__   .    ,     _ 
* 


T/r 

—  =COS^=-7=, 

Vq  Vr 

which  may  be  used  to  compute  q,  v,  and  vff  when  v"  —  v  is  known. 

When  \(v"  —  v)  and  J(v"  +  v),  and  hence  t>"  and  v,  have  been 
determined,  the  time  of  perihelion  passage  must  be  found,  as  already 
explained,  by  means  of  Table  VI.  or  Table  VIII. 

It  is  evident,  therefore,  that  in  the  determination  of  an  orbit,  as 
soon  as  the  numerical  values  of  r,  r",  and  Y.  have  been  derived  from 
the  equations  (52),  instead  of  completing  the  calculation  of  the  ele- 
ments of  the  orbit,  we  may  find  q  and  T,  and  then,  by  means  of 
these,  the  values  of  rf  and  vf  may  be  computed  directly.  When  this 
has  been  effected,  the  values  of  n  and  n"  may  be  found  from  (3),  or 

that  of  —  ,  from  (34).     Then  we  compute  p  by  means  of  the  first  of 

equations  (70),  and  the  corrected  value  of  M  from  (33),  or,  in  the 
special  cases  already  examined,  from  the  equations  (37)  and  (39).  In 
this  way,  by  successive  approximations,  the  determination  of  para- 
bolic elements  from  given  data  may  be  carried  to  the  limit  of  accuracy 
which  is  consistent  with  the  assumption  of  parabolic  motion.  In  the 
case,  however,  of  the  equations  (37)  and  (39),  the  neglected  terms 
may  be  of  the  second  order,  and,  consequently,  for  the  final  results 
it  will  be  necessary,  in  order  to  attain  the  greatest  possible  accuracy, 
to  derive 

M=?"- 

P 

from  (15)  and  (16).  When  the  final  value  of  M  has  been  found,  the 
determination  of  the  elements  is  completed  by  means  of  the  formula; 
already  given. 


PARABOLIC   ORBIT.  199 

72.  EXAMPLE. — To  illustrate  the  application  of  the  formulae  for 
the  calculation  of  the  parabolic  elements  of  the  orbit  of  a  comet  by 
a  numerical  example,  let  us  take  the  following  observations  of  the 
Fifth  Comet  of  1863,  made  at  Ann  Arbor: — 

Ann  Arbor  M.  T.                                 a  6 

1864  Jan.  10  6*  57m  20-.5  19*  Um  4'.92  +  34°    6'  27".4, 

13  6   11    54 .7  19   25    2 .84  36    36  52  .8, 

16  6   35    11 .6  19  41    4  .54  +  39    41  26  .9. 

These  places  are  referred  to  the  apparent  equinox  of  the  date  and 
are  already  corrected  for  parallax  and  aberration  by  means  of 
approximate  values  of  the  geocentric  distances  of  the  comet.  But 
if  approximate  values  of  these  distances  are  not  already  known,  the 
corrections  for  parallax  and  aberration  may  be  neglected  in  the  first 
determination  of  the  approximate  elements  of  the  unknown  orbit  of 
a  comet.  If  we  convert  the  observed  right  ascensions  and  declina- 
tions into  the  corresponding  longitudes  and  latitudes  by  means  of 
equations  (1),  and  reduce  the  times  of  observation  to  the  meridian 
of  Washington,  we  get 

Washington  M.  T.                                 A  (3 

1864  Jan.  10  7ft  24"    3'  297°  53'    7".6  -f  55°  46'  58".4, 

13  6   38    37  302    57  51  .3  57    39  35  .9, 

16  7     1    54  310    31  52  .3  +  59    38  18  .7. 

Next,  we  reduce  these  places  by  applying  the  corrections  for  pre- 
cession and  nutation  to  the  mean  equinox  of  1864.0,  and  reduce  th<* 
times  of  observation  to  decimals  of  a  day,  and  we  have 

t  =  10.30837,  A  =  297°  52'  51".l,  0  =  +  55°  46'  58".4, 
1?  =  13.27682,  A'  =  302  57  34  .4,  /?'  =  57  39  35  .9, 
tf'  =  16.29299,  A"  =  310  31  35  .0,  £"=  +  59  38  18  .7. 

For  the  same  times  we  find,  from  the  American  Nautical  Almanae, 

O    =290°    6' 27".4,  log  E  =9.992763, 

0'  =293      7  57  .1,  logJ?'  =9.992830, 

O"  =  296    12  15  .7,  log  #'  =  9.992916, 

which  are  referred  to  the  mean  equinox  of  1864.0.  It  will  gene- 
rally be  sufficient,  in  a  first  approximation,  to  use  logarithms  of  five 
decimals ;  but,  in  order  to  exhibit  the  calculation  in  a  more  complete 
form,  we  shall  retain  six  places  of  decimals. 

Since  the  intervals  are  very  nearly  equal,  we  may  assume 


200  THEORETICAL   ASTRONOMY. 


- 

n"~r"~  N"' 
Then  we  have 

M==t"-  t      tan/5'sin(A—  O')  —  tan/9sin(A'—  O') 


H  —  t    tan  ?'  sin  (A'  —  0  ')  —  tan  jf  sin  (A"  —  O')' 
and 

g  sin  (  £  -  O)  =  R"  sin  (0"  -  Q), 
g  cos(G  —  O)  =  R"  cos(O"  —  O)  —  R; 
h  cos  C  cos  (H—  A")  =  Jf  _  cos  (X"  —  A), 
A  cos  C  sin  (jff  —  A")  =  sin  (A"  —  A), 
AsinC  =  M  tan/5"  —  tan/3; 

from  which  to  find  Jf,  6r,  #,  IT,  C>  and  ^     Thus  we  obtain 

log  M=  9.829827,  JET=r      94°  24'    1".8, 

G  =  22°  58'  1".7,  C  =  —  40    28  21  .9, 

log  g  =  9.019613,  log  h  =  9.688532. 

A"  cos  /? 

Since  —  r-  =  M  --  -^  =  0.752.  it  appears  that  the  comet,  at  the  time 
A  cos  p 

of  these  observations,  was  rapidly  approaching  the  earth.  The 
quadrants  in  which  G  —  O  and  H  —  k"  must  be  taken,  are  deter- 
mined by  the  condition  that  g  and  h  cos  £  must  always  be  positive. 
The  value  of  M  should  be  checked  by  duplicate  calculation,  since  an 
error  in  this  will  not  be  exhibited  until  the  values  of  Xf  and  f}f  are 
computed  from  the  resulting  elements. 
Next,  from 

cos  4.  =  cos  /5  cos  (A  —  O),  cos  t"  =  cos  /9"  cos  (A"  —  0";, 

cos  <p  =  cos  C  cos  (  G  —  -ET), 

we  compute  cos  oj/,  cos  ij/',  and  cos  <p  •  and  then  from 

g  sin  <p      =  A,  h  cos  /?  =  b, 


<7  cos  <p  —  bR  cos  4»  =  c,  g  cosy  —  6"jR"  cos  4"  =  c'', 

we  obtain  J.,  jB,  J5/r,  &c.  It  will  generally  be  sufficiently  exact  to 
find  sin  ^  and  sin  ty'  from  cos  i/,  and  cos  4//r  ;  but  if  more  accurate 
values  of  ^  and  ^/"  are  required,  they  may  be  obtained  by  means  of 
the  equations  (42)  and  (43).  Thus  we  derive 

log  A  =  9.006485,        log  P  =  9.912052,        log  B"  =  9.933366, 
log  b  =  9.438524,  log  b"  =  9.562387, 

c  ^  —  0.125067,  c"  =  —  0.150562. 


NUMERICAL   EXAMPLE.  201 

Then  we  have 


2r' 


from  which  to  find,  by  successive  trials,  the  values  of  r,  r",  and  X, 
that  of  [J.  being  found  from  Table  XL  with  the  argument  57.  First, 
we  assume 

log  x  =  log  (Y  1/2)  =  9.163132, 

and  with  this  we  obtain 
log  r  =  9.913895,        log  r"  =  9.938040,        log  (r  +  r")  =  0.227165. 

This  value  of  log(r  +  r"}  gives  7  =  0.094,  and  from  Table  XL  we 
find  log  //  =  0.000160.  Hence  we  derive 

log  x  =  9.200220,        log  r  =  9.912097,        log  r"  =  9.935187, 
log  (r  +  r")  =  0.224825. 

Repeating  the  operation,  using  the  last  value  of  log(r  -f  r"),  we  get 

log  x  =  9.201396,        log  r  =  9.912083,        log  r"  =  9.935117, 
log  (r  +  r")  =  0.224783. 

The  correct  value  of  log  (r  -f  r")  may  now  be  found  by  means  of  the 
equation  (67).  Thus,  we  have,  in  units  of  the  sixth  decimal  place  of 
the  logarithms, 

a  =  224825  —  227165  =  —  2340,        a'  =  224783  —  224825  =  —  42, 
and  the  correction  to  the  last  value  of  log(r  +  r")  becomes 


a  —  a 
Therefore, 

log  (r  +  t")  =  0.224782, 

and,  recomputing  ^,  //,  x,  r,  and  r",  we  get,  finally, 

log  x  =  9.201419,        logr  =  9.912083,        log/'  =  9.935116, 
log  (r  +  r")  =  0.224782. 

The  agreement  of  the  last  value  of  log(r  +  r"}  with  the  preceding 
one  shows  that  the  results  are  correct.     Further,  it  appears  from  the 


202  THEORETICAL   AS1KOXOMY. 

values  of  r  and  r"  that  the  comet  had  passed  its  perihelion  and  was 
receding  from  the  sun. 

By  means  of  the  values  of  r  and  r"  we  might  compute  approxi- 

/y?*' 

mate  values  of  r'  and  -^-  from  the  equations  (30)  and  (31),  and  then 
dt  ._r 

n  _/V 

a  more  approximate  value  of  —j-,  from  (28),  that  of  ^  being  found 

from  (32).     But,  since  rf  differs  but  little  from  Rf,  the  difference 

n  N 

between  —77  and  -^  is  very  small,  so  that  it  is  not  necessary  to  con- 

sider the  second  term  of  the  second  member  of  the  equation  (33); 
and,  since  the  intervals  are  very  nearly  equal,  the  error  of  the  as- 
sumption 

n         r 


is  of  the  third  order.  It  should  be  observed,  however,  that  an  error 
in  the  value  of  M  affects  H}  £,  A,  and  hence  also  A,  6,  6",  c,  and  c", 
and  the  resulting  value  of  p  may  be  affected  by  an  error  which  con- 
siderably exceeds  that  of  M.  It  is  advantageous,  therefore,  to  select 
observations  which  furnish  intervals  as  nearly  equal  as  possible  in 
order  that  the  error  of  M  may  be  small,  otherwise  it  may  become 
necessary  to  correct  If  and  to  repeat  the  calculation  of  r,  r"9  and  x. 
We  may  also  compute  the  perihelion  distance  and  the  time  of  peri- 
helion passage  from  r,  rn  ',  and  x  by  means  of  the  equations  (86),  (89), 
and  (91)  in  connection  with  Tables  VI.  and  VIII.  Then  rf  and  vf 
may  be  computed  directly,  and  the  complete  expression  for  M  may 
be  employed. 

In  the  first  determination  of  the  elements,  and  especially  when  the 
corrections  for  parallax  and  aberration  have  been  neglected,  it  is  un- 
necessary to  attempt  to  arrive  at  the  limit  of  accuracy  attainable, 
since,  when  approximate  elements  have  been  found,  the  observations 
may  be  more  conveniently  reduced,  and  those  which  include  a  longer 
interval  may  be  used  in  a  more  complete  calculation.  Hence,  as  soon 
as  r,  rn  ',  and  x  have  been  found,  the  curtate  distances  are  next  deter 
mined,  and  then  the  elements  of  the  orbit.  To  find  p  and  prf,  we 

have 

d=  +  0.122395, 

the  positive  sign  being  used  since  7,  is  greater  than  g,  and  the  formulae 


gve 

log  p  =  9.480952,  log  P"  =  9.310779. 


NUMERICAL   EXAMPLE.  203 

From  these  values  of  p  and  //',  it  appears  that  the  comet  was  very 
near  the  earth  at  the  time  of  the  observations. 

The  heliocentric  places  are  then  found  by  means  of  the  equations 
(71)  and  (72).  Thus  we  obtain 

I  =  106°  40'  50".5,        b  =  -f  33°    1'  10".6,        logr  =  9.912082, 
r=112    31     9.9,        b"  =  +  23    55     5.8,        log  r"=  9.935116. 

The  agreement  of  these  values  of  r  and  rn  with  those  previously 
found,  checks  the  accuracy  of  the  calculation.  Further,  since  the 
heliocentric  longitudes  are  increasing,  the  motion  is  direct. 

The  longitude  of  the  ascending  node  and  the  inclination  of  the 
orbit  may  now  be  found  by  means  of  the  equations  (74),  (75),  or  (70)  ; 
and  we  get 

£  =  304°  43'  11".5,  %  =  64°  31'  21".7. 

The  values  of  u  and  uff  are  given  by  the  formulae 


COS  I  COS  I 

u  and  I  —  &  being  in  the  same  quadrant  in  the  case  of  direct  motion, 
Thus  we  obtain 

u  —  142°  52'  12".4,  u"=  153°  18'  49".4. 

Then  the  equation 

x*  =  (r"  —  r  cos  (u"  —  u))*  +  r2  sin2  (u"  —  u} 
gives 

log  x  =  9.201423, 

and  the  agreement  of  this  value  of  K  with  that  previously  found, 
proves  the  calculation  of  &,  i,  u,  and  u". 
From  the  equations 

tan  (45°  -f  0')  =  ^/~, 

1      .    .  1  1  ,  ,;        v          ,  tan  20' 

—j=  sin  i  (i  (u  4-  u)  —  w)  =  -         —  —  —  , 

/  <i   \2    ^  I  /  J  t     f       it  •*.     4  /  -  Tf' 

Vq  sin  \  (u  —  tt)  i/rr 

1  >«  /  jy  i     \         \  sec  20' 

(I  (t*"  -f-  «)  —  «,)  ±=  -  -—  -  -j—  =, 


cos  I  (t*  —  w)  K  rr 
we  get 

0'  =  0°  22'  47".4,        w  =  115°  40'  6';.3,        log  q  =  9.887378. 

Hence  we  have 

*  =  01  -f  £  =  60°  23'  17".  S, 


204  THEORETICAL   ASTRONOMY. 

and 

i  =  u  —  «>  =  27°  12'  6".l,  v"  =  ti"  —  «  =  37°  38'  43  '.1. 

Then  we  obtain 

log  m  =  9.9601277  —  |  log  5  =  0.129061, 
and,  corresponding  to  the  values  of  v  and  v",  Table  VI.  gives 

log  M  =  1.267163,  log  M"  =  1.424152. 

Therefore,  for  the  time  of  perihelion  passage,  we  have 


and 


T=t  ——  =  t  —  13.74364, 
m 

T=t'——  =f—  19.72836. 
m 


The  first  value  gives  T=  1863  Dec.  27.56473,  and  the  second  gives 
T=  Dec.  27.56463.  The  agreement  between  these  results  is  the  final 
proof  of  the  calculation  of  the  elements  from  the  adopted  value  of 

M=f-. 

P 

If  we  find  T  by  means  of  Table  VIII.,  we  have 

log  N  =  0.021616,  log  N"  =  0.018210, 

and  the  equation 

2  2 

T  =  t  —  3£  Nr%  sin  v  =  if'  —  3^  N"i"*  sin  v", 

in  which  log  ^  =  1.5883273,  gives  for  T  the  values  Dec.  27.56473 

and  Dec.  27.56469. 

Collecting  together  the  several  results  obtained,  we  have  the  fol- 
lowing elements: 

T  —  1863  Dec.  27.56471  Washington  mean  time. 

=   60°23'17".8       _._    .        ,  „ 

ic  and  Mean 


log  q  =  9.887378. 

Motion  Direct. 

73.  The  elements  thus  derived  will,  in  all  cases,  exactly  represent 
the  extreme  places  of  the  comet,  since  these  only  have  been  used  in 
finding  the  elements  after  p  and  p"  have  been  found.  If,  by  means 


NUMERICAL   EXAMPLES.  205 

of  these  elements,  we  compute  n  and  nn ',  and  correct  the  \alue  of  M, 
the  elements  which  will  then  be  obtained  will  approximate  nearer 
the  true  values;  and  each  successive  correction  will  furnish  more 
accurate  results.  When  the  adopted  value  of  M  is  exact,  the  result- 
ing elements  must  by  calculation  reproduce  this  value,  and  since  the 
computed  values  of  A,  A",  /9,  and  ft"  will  be  the  same  as  the  observed 
values,  the  computed  values  of  X  and  ft'  must  be  such  that  when 
substituted  in  the  equation  for  M,  the  same  result  will  be  obtained 
as  when  the  observed  values  of  X'  and  ft'  are  used.  But,  according 
to  the  equations  (13)  and  (14),  the  value  of  M  depends  only  on  the 
inclination  to  the  ecliptic  of  a  great  circle  passing  through  the  places 
of  the  sun  and  comet  for  the  time  £',  and  is  independent  of  the  angle 
at  the  earth  between  the  sun  and  comet.  Hence,  the  spherical  co- 
ordinates of  any  point  of  the  great  circle  joining  these  places  of  the 
sun  and  comet  wilj,  in  connection  with  those  of  the  extreme  places, 
give  the  same  value  of  M,  and  when  the  exact  value  of  M  has  been 
used  in  deriving  the  elements,  the  computed  values  of  A'  and  ft'  must 
give  the  same  value  for  wr  as  that  which  is  obtained  from  observa- 
tion. But  if  we  represent  by  if/  the  angle  at  the  earth  between  the 
sun  and  comet  at  the  time  £',  the  values  of  if/  derived  by  observation 
and  by  computation  from  the  elements  will  differ,  unless  the  middle 
place  is  exactly  represented.  In  general,  this  difference  will  be  small, 
and  since  w'  is  constant,  the  equations 

cos  4/  =  cos  p  cos  (A'  —  O')> 

sin  4-'  cos  w'  =  cos  ft'  sin  (A'  —  ©'),  (93) 

sin  4/  sin  w'  =  sin  ft, 

give,  by  differentiation, 

cos  p  dl'  =  cos  w'  sec  p  d$r, 

dp  =  smufcoa(X  —  O'HV- 
From  these  we  get 

cosft'dA'       tan(*'— Q') 
dp  sin  p 

which  expresses  the  ratio  of  the  residual  errors  in  longitude  and 
latitude,  for  the  middle  place,  when  the  correct  value  of  M  has  been 
used. 

Whenever  these  conditions  are  satisfied,  the  elements  will  be 
correct  on  the  hypothesis  of  parabolic  motion,  and  the  magnitude 
of  the  final  residuals  in  the  middle  place  will  depend  on  the  deviation 
of  the  actual  orbit  of  the  comet  from  the  parabolic  form.  Further, 


206  THEORETICAL   ASTRONOMY. 

when  elements  have  been  derived  from  a  value  of  M  which  has  not 
been  finally  corrected,  if  we  compute  Xr  and  $'  by  means  of  these 
elements,  and  then 


the  comparison  of  this  value  of  tan  wf  with  that  given  by  observa- 
tion will  show  whether  any  further  correction  of  M  is  necessary,  and 
if  the  difference  is  riot  greater  than  what  may  be  due  to  unavoidable 
errors  of  calculation,  we  may  regard  M  as  exact. 

To  compare  the  elements  obtained  in  the  case  of  the  example 
given  with  the  middle  place,  we  find 

vf  =  32°  31'  13".5,  u'  =  148°  IV  19".8,  log  /  ==  9.922836. 

Then  from  the  equations 

tan  (f  —  Q>  )  =  cos  i  tan  u', 

tan  bf  =  tan  i  sin  (lf  —  &  ), 
we  derive 

I'  =  109°  46'  48".3,  V  =  28°  24'  56".0. 

By  means  of  these  and  the  values  of  O'  and  Rf,  we  obtain 
A'  =  302°  57'  41".l,  p  =  57°  39'  37".0  ; 

and,  comparing  these  results  with  the  observed  values  of  X'  and  /?', 
the  residuals  for  the  middle  place  are  found  to  be 

Comp.  —  Obs. 
cos  p  AA'  =  -f  3".6,  A/9  =  +  I'M. 

The  ratio  of  these  remaining  errors,  after  making  due  allowance  for 
unavoidable  errors  of  calculation,  shows  that  the  adopted  value  of 
M.  is  not  exact,  since  the  error  of  the  longitude  should  be  less  than 
that  of  the  latitude. 

The  value  of  w1  given  by  observation  is 

log  tan  w'  =  0.966314, 

and  that  given  by  the  computed  values  of  A'  and  ft'  is 
log  tan  w'  =  0.966247. 

The  difference  being  greater  than  what  can  be  attributed  to  errors  of 
calculation,  it  appears  that  the  value  of  M  requires  further  cor- 


NUMERICAL   EXAMPLES.  207 

rection.  Since  the  difference  is  small,  we  may  derive  the  correct 
value  of  M  by  using  the  same  assumed  value  of  — ,,  and,  instead  of 

Yb 

the  value  of  tan&/  derived  from  observation,  a  value  differing  as 
much  from  this  in  a  contrary  direction  as  the  computed  value  differs. 
Thus,  in  the  present  example,  the  computed  value  of  log  tan  wf  is 
0.000067  less  than  the  observed  value,  and,  in  finding  the  new  value 

of  Mj  we  must  use 

log  tan  w'  =  0.966381 

in  computing  /?0  and  /?/'  involved  in  the  first  of  equations  (14).  If 
the  first  of  equations  (10)  is  employed,  we  must  use,  instead  of  tan/9' 
as  derived  from  observation, 

tan  p  =  tan  vf  sin  (A'  —  0'), 
or 

log  tan  f  =  0.966381  +  log  sin  (A'  —  0')  =  0.198559, 

the  observed  value  of  X'  being  retained.     Thus  we  derive 

log  M=  9.829586, 

and  if  the  elements  of  the  orbit  are  computed  by  means  of  this 
value,  they  will  represent  the  middle  place  in  accordance  with  the 
condition  that  the  difference  between  the  computed  and  the  observed 
value  of  tan  wr  shall  be  zero. 

A    system  of   elements    computed    with    the    same    data    from 
log  M=  9.822906  gives  for  the  error  of  the  middle  place, 

C.-O. 

cos  f  A;/  =  —  1'  26".2,  &p  =  —  40".l. 

If  we  interpolate  by  means  of  the  residuals  thus  found  for  two  values 
of  M,  it  appears  that  a  system  of  elements  computed  from 

log  M=  9.829586 

will  almost  exactly  represent  the  middle  place,  so  that  the  data  are 
completely  satisfied  by  the  hypothesis  of  parabolic  motion. 
The  equations  (34)  and  (32)  give 

log  -^-  =  0.006955,  log  -^  =  0.006831, 

/f\/  JLV 

and  from  (10)  we  get 

log  M '  =  9.822906,  log  M"  =  9.663729... 


208  THEORETICAL   ASTRONOMY. 

Then  by  means  of  the  equation  (33)  we  derive,  for  the  corrected 

value  of  My 

log  M=  9.829582, 

which  differs  only  in  the  sixth  decimal  place  from  the  result  obtained 
by  varying  tanw'  and  retaining  the  approximate  values  —t  =  -rf  =  j™' 

74.  When  the  approximate  elements  of  the  orbit  of  a  comet  are 
known,  they  may  be  corrected  by  using  observations  which  include 
a  longer  interval  of  time.  The  most  convenient  method  of  effecting 
this  correction  is  by  the  variation  of  the  geocentric  distance  for  the 
time  of  one  of  the  extreme  observations,  and  the  formulae  which 
may  be  derived  for  this  purpose  are  applicable,  without  modification, 
to  any  case  in  which  it  is  possible  to  determine  the  elements  of  the 
orbit  of  a  comet  on  the  supposition  of  motion  in  a  parabola.  Since 
there  are  only  five  elements  to  be  determined  in  the  case  of  parabolic 
motion,  if  the  distance  of  the  comet  from  the  earth  corresponding  to 
the  time  of  one  complete  observation  is  known,  one  additional  com- 
plete observation  will  enable  us  to  find  the  elements  of  the  orbit. 
Therefore,  if  the  elements  are  computed  which  result  from  two  or 
more  assumed  values  of  A  differing  but  little  from  the  correct  value, 
by  comparison  of  intermediate  observations  with  these  different  sys- 
tems of  elements,  we  may  derive  that  value  of  the  geocentric  distance 
of  the  comet  for  which  the  resulting  elements  will  best  represent  the 
observations. 

In  order  that  the  formulae  may  be  applicable  to  the  case  of  any 
fundamental  plane,  let  us  consider  the  equator  as  this  plane,  and, 
supposing  the  data  to  be  three  complete  observations,  let  A,  Af,  A" 
be  the  right  ascensions,  and  D,  D',  D"  the  declinations  of  the  sun 
for  the  times  t,  t',  t".  The  co-ordinates  of  the  first  place  of  the  earth 
referred  to  the  third  are 

x  —  R"  cos  D"  cos  A"  —  R  cos  D  cos  A, 
y  =  R"  cos  D"  sin  A"  —  EcosD  sin  A, 
z=R"smD"  —  EsmD. 

If  we  represent  by  g  the  chord  of  the  earth's  orbit  between  the  places 
for  the  first  and  third  observations,  and  by  G  and  K,  respectively, 
the  right  ascension  and  declination  of  the  first  place  of  the  earth  as 
seen  from  the  third,  we  shall  have 

x  =  g  cos  K  cos  G, 
y  =  g  cos  K  sin  G, 
z  =  g  sin  K. 


VAKIATION   OF   THE   GEOCENTKIC   DISTANCE.  209 

and,  consequently, 

g  cos  K  cos  (  G  —  A)  =  R"  cos  D"  cos  (A"  —  A)  —  R  cos  D, 

g  cos  K  sin  (  G  —  A)  =  jR"  cos  D"  sin  (A"  —  A),  (96) 

g  sin  iT  =  .#"  sin  Z>"  —  jR  sin  D, 

from  which  #,  JT,  and  G  may  be  found. 

If  we  designate  by  xn  yn  z,  the  co-ordinates  of  the  first  place  of 
the  comet  referred  to  the  third  place  of  the  earth,  we  shall  havf» 

x,  =  A  cos  5  cos  a  -f  g  cos  K  cos  G, 
y,  —  A  cos  d  sin  a  -f-  g  cos  K  sin  (r, 
2,  =  A  sin  5  -f-  #  sin  K. 

Let  us  now  put 

x,  =  h'  cos  C'  cos  IT, 
^  ==  A'  cos  :'  sin  #', 
z,  =  h'  sin  C', 
and  we  get 

hf  cos  %  cos  (.ff '  —  G)  =  A  cos  fl  cos  (a  —  G)  +  0  cos  JT, 

A'  cos  C'  sin  GET  —  G)  =  J  cos  «  sin  (a  —  (f).  C97) 

A.'  sin  C'  =  J  sin  <S  -j-  <7  sm  -^ 

from  which  to  determine  H',  £',  and  hf. 

If  we  represent  by  <pr  the  angle  at  the  third  place  of  the  earth 
between  the  actual  first  and  third  places  of  the  comet  in  space,  we 
obtain 

cos  <f>'=  cos  C'  cos  H'  cos  <5"  cos  a"-\-  cos  C'  sin  JET'  cos  5"  sin  a"-}-  sin  C'  sin  <5", 

or 

cos  ?/  =  cos  £'  cos  <T  cos  (a"  —  H')  +  sin  C'  sin  <5" ;  (98) 

and  if  we  put 

e  sin/=sin<5", 

e  cosf=  cos  d"  cos  (a"  —  H') 
this  becomes 

cos  ?'  =  e  cos  (C'  — /).  (99) 

Then  we  shall  have 

x2  =  /t'2+  J"2—  2V J"  cos?/ 
or 

X2  =  ( j"  _  tf  CoS  ?')» -f  h"  sin2  ?',  (1 00) 

in  which  A"  is  the  distance  of  the  comet  from  the  earth  coi  respond- 
ing to  the  last  observation.     We  have,  also,  from  equations  (44)  and 

(45), 

r2   =(J   —  .Rcos*)2     -f-JR2sin24, 
r'"  =  ( A"  —  R"  cos  4/')2  +  R"*  sin1  V', 

14 


210  THEORETICAL,   ASTRONOMY. 

in  which  ^  is  the  angle  at  the  earth  between  the  sun  and  comet  at 
the  time  t,  and  oj/'  the  same  angle  at  the  time  t"  .  To  find  their 
values,  we  have 

cos  4*  =  cos  D  cos  d  cos  (a  —  A)  -f-  sin  D  sin  <5,  ,.  ~~\ 

cos  *"=  cos  Z)"  cos  *"  cos  (a"—  A"}  +  sin  D"  sin  5", 

which  may  be  still  further  reduced  by  the  introduction  of  auxiliary 
angles  as  in  the  case  of  equation  (98). 
Let  us  now  put 

hf  sin  <?'  =  C,  h'  cos  <f>'  =  c, 

Rsm*=B,  R  cos  *  =  b,  (103) 

R"  sin  4,"  =  £",  R"  cos  4"  =  &", 

and  we  shall  have 


=  T/(J"  —  c)2-f  C2, 

&)2  +  £2,  (104) 


'These  equations,  together  with  (56),  will  enable  us  to  determine  J" 
by  successive  trials  when  J  is  given. 

We  may,  therefore,  assume  an  approximate  value  of  A"  by  means 
of  the  approximate  elements  known,  and  find  r"  from  the  last  of 
these  equations,  the  value  of  T  having  been  already  found  from  the 
assumed  value  of  J.  Then  x  is  obtained  from  the  equation 

2r' 


fj.  being  found  by  means  of  Table  XI.,  and  a  second  approximation 
to  the  value  of  J"  from 


*.  (105) 

The  approximate  elements  will  give  A"  near  enough  to  show  whether 
the  upper  or  lower  sign  must  be  used.  With  the  value  of  A"  thus 
found  we  recompute  r"  and  x  as  before,  and  in  a  similar  manner  find 
a  still  closer  approximation  to  the  correct  value  of  An  '.  A  few  trial? 
will  generally  give  the  correct  result. 

When  A"  has  thus  been  determined,  the  heliocentric  places  are 
found  by  means  of  the  formulae 

r  cos  b  cos  (I  —  A)  =  A  cos  d  cos  (a  —  A)  —  E  cos  D, 
r  cos  b  sin  (/  —  A)  =  A  cos  d  sin  (a  —  A), 
r  sin  b  =  A  sin  <5  —  R  sin  D  ; 


VARIATION   OF   THE   GEOCENTRIC   DISTANCE.  211 

r"  cos  V  cos  <7'  —  A")  =  A"  cos  V  cos  (a"  -  A")  —  K"  cos  £", 

r"  cos  6"  sin  (r  —  A"}  =  A"  cos  d"  sin  (a"  —  4"),  (107) 

r"  sin  6"  =  J"  sin  5"  —  £"  sin  Z>", 

in  which  6,  b"y  I,  I"  are  the  heliocentric  spherical  co-ordinates  re- 
ferred to  the  equator  as  the  fundamental  plane.  The  values  of  r  and 
r"  found  from  these  equations  must  agree  with  those  obtained  from 
(104). 

The  elements  of  the  orbit  may  now  be  determined  by  means  of  the 
equations  (75),  (77),  and  (81),  in  connection  with  Tables  VI.  and 
VIII.,  as  already  explained.  The  elements  thus  derived  will  be  re- 
ferred to  the  equator,  or  to  a  plane  passing  through  the  centre  of  the 
sun  and  parallel  to  the  earth's  equator,  and  they  may  be  transformed 
into  those  for  the  ecliptic  as  the  fundamental  plane  by  means  of  the 
equations  (109)^ 

75.  With  the  resulting  elements  we  compute  the  place  of  the  comet 
for  the  time  tf  and  compare  it  with  the  corresponding  observed  place, 
and  if  we  denote  the  computed  right  ascension  and  declination  by  a/ 
and  dQ',  respectively,  we  shall  have 

a'  +  <*'=»„',  *+*  =  ».', 

in  which  a'  and  d'  denote  the  differences  between  computation  and 
observation.  Next  we  assume  a  second  value  of  J,  which  we  repre- 
sent by  J  -f-  d  J,  and  compute  the  corresponding  system  of  elements. 
Then  we  have 


a!'  and  d"  denoting  the  differences  between  computation  and  obser- 
vation for  the  second  system  of  elements.  We  also  compute  a  third 
system  of  elements  with  the  distance  J  —  £J,  and  denote  the  differ- 
ences between  computation  and  observation  by  a  and  d;  then  we  shall 
have 


and  similarly  for  c?,  df,  and  d".     If  these  three  numbers  are  exactly 
represented  by  the  expression 


m 


in  which  J  +  x  is  the  general  value  of  the  argument,  since  the  values 
}f  a,  a',  and  a"  will  be  such  that  the  third  differences  may  be  neg- 
lected, this  formula  may  be  assumed  to  express  exactly  any  value  of 
the  function  corresponding  to  a  value  of  the  argument  not  differing 


212  THEORETICAL   ASTRONOMY. 

much  from  J,  or  within  the  limits  x  =  —  dA  and  x  =  -f-  <5J,  the  as- 
sumed values  J  —  d  J,  J,  and  J  +  £  J  being  so  taken  that  the  correct 
value  of  J  shall  be  either  within  these  limits  or  very  nearly  so. 
To  find  the  coefficients  m,  n,  and  o,  we  have 

m  —  n  -f-  o  =  a.  m  —  a',  m  -{-  n-\-  o  —  a", 

whence 

?»  =  a',  n  =  £(a"—  a),  o  =  £  (a"  -f  a)  —  a'. 

Now,  in  order  that  the  middle  place  may  be  exactly  represented  in 
right  ascension,  we  must  have 


which  we  find 


or 


In  the  same  manner,  the  condition  that  the  middle  place  shall  be 
exactly  represented  in  declination,  gives 


In  order  that  the  orbit  shall  exactly  represent  the  middle  place,  both 
conditions  must  be  satisfied  simultaneously;  but  it  will  rarely  happen 
that  this  can  be  effected,  and  the  correct  value  of  x  must  be  found 
from  those  obtained  by  the  separate  conditions.  The  arithmetical 
mean  of  the  two  values  of  x  will  not  make  the  sum  of  the  squares 
of  the  residuals  a  minimum,  and,  therefore,  give  the  most  probable 
value,  unless  the  variation  of  cos  3f  AO/,  for  a  given  increment  as- 
signed to  J,  is  the  same  as  that  of  &d'.  But  if  we  denote  the  value 
of  x  for  which  the  error  in  a'  is  reduced  to  zero  by  x',  and  that  for 
which  &d'  =  0,  by  x",  the  most  probable  value  of  x  will  be 

•-•- 


in  which  n  =  $(aff  —  a)  and  nf  =  \(dff  --  d).  It  should  be  observed 
that,  in  order  that  the  differences  in  right  ascension  and  declination 
shall  have  equal  influence  in  determining  the  value  of  x9  the  values 
of  a,  a',  and  a"  must  be  multiplied  by  cos  8f.  The  value  of  dA  is 
most  conveniently  expressed  in  units  of  the  last  decimal  place  of  the 
logarithms  employed. 


NUMERICAL   EXAMPLE.  213 

If  the  elements  are  already  known  so  approximately  that  the  first 
assumed  value  of  J  differs  so  little  from  the  true  value  that  the 
second  differences  of  the  residuals  may  be  neglected,  two  assumptions 
in  regard  to  the  value  of  A  will  suffice.  Then  we  shall  have  o  =  0, 
and  hence 

m  =  af,  n  =  a"  —  a'. 

The  condition  that  the  middle  place  shall  be  exactly  represented, 
gives  the  two  equations 

"'  '  0, 

Q. 

The  combination  of  these  equations  according  to  the  method  of  least 
squares  will  give  the  most  probable  value  of  x,  namely,  that  for 
which  the  sum  of  the  squares  of  the  residuals  will  be  a  minimum. 

Having  thus  determined  the  most  probable  value  of  #,  a  final 
system  of  elements  computed  with  the  geocentric  distance  A  -f-  #, 
corresponding  to  the  time  t,  will  represent  the  extreme  places  exactly, 
and  will  give  the  least  residuals  in  the  middle  place  consistent  with 
the  supposition  of  parabolic  motion.  It  is  further  evident  that  we 
may  use  any  number  of  intermediate  places  to  correct  the  assumed 
value  of  J,  each  of  which  will  furnish  two  equations  of  condition 
for  the  determination  of  x,  and  thus  the  elements  may  be  found 
which  will  represent  a  series  of  observations. 

76.  EXAMPLE. — The  formulae  thus  derived  for  the  correction  of 
approximate  parabolic  elements  by  varying  the  geocentric  distance, 
are  applicable  to  the  case  of  any  fundamental  plane,  provided  that 
a,  d,  A,  D,  &c.  have  the  same  signification  with  respect  to  this  plane 
that  they  have  in  reference  to  the  equator.  To  illustrate  their 
numerical  application,  let  us  take  the  following  normal  places  of 
the  Great  Comet  of  1858,  which  were  derived  by  comparing  an 
ephemeris  with  several  observations  made  during  a  few  days  before 
and  after  the  date  of  each  normal,  and  finding  the  mean  difference 
between  computation  and  observation : 


Washington  M.  T. 
1858  June  11.0 
July  13.0 
Aug.  14.0 

a 

141°  18'  30".9 
144    32  49  .7 
152    14  12  .0 

6 

+  24°  46'  25".4, 
27    48    0  .8, 
-f  31    21  47  .9, 

which  are  referred  to  the  apparent   equinox  of  the   date.     These 
places  are  free  from  aberration. 


214  THEOEETICAL   ASTRONOMY. 

We  shall  take  the  ecliptic  for  the  fundamental  plane,  and  con- 
verting these  right  ascensions  and  declinations  into  longitudes  and 
latitudes,  and  reducing  to  the  ecliptic  and  mean  equinox  of  1858.0, 
the  times  of  observation  being  expressed  in  days  from  the  beginning 
of  the  ye^tr,  we  get 

t  =  162.0,  A  =  135°  51'  44".2,  p  =  +    9°    6'  57".8, 

If  =  194.0,  *'  =  137    39  41  .2,  p  =      12    55     9  .0, 

*"  =  226.0,  A"  =  142    51  31  .8,  ^'  =  +  18    36  28  .7. 

From  the  American  Nautical  Almanac  we  obtain,  for  the  true  places 
of  the  sun, 

Q   =   80°  24'  32".4,  logJ?  =0.006774, 

O'  =110    55  51  .2,  log^R'  =0.007101, 

0"  =  141    33     2  .0,  log  R"  =  0.005405, 

the  longitudes  being  referred  to  the  mean  equinox  1858.0. 

When  the  ecliptic  is  the  fundamental  plane,  we  have,  neglecting 
the  sun's  latitude,  D  •=  0,  and  we  must  write  ^  and  ft  in  place  of  a 
and  <5,  and  O  in  place  of  A,  in  the  equations  which  have  been  derived 
for  the  equator  as  the  fundamental  plane.  Therefore,  we  have 

g  cos  (Q  —  O)  =  R"  cos  (O"  —  O)  —  R, 
<7  sin  (£  -  0)  =  R"  sin  (O"  -  O)  ; 

cos  4  =  cos  p  cos  (A  —  O),  cos  4/'  =  cos  /3"  cos  (A"  —  0") 

R  cos  4  =  b,  12"  cos  *"  =  &", 

JB,  12"sin4/'  =  .B", 


from  which  to  find  G,  g,  b,  B,  b",  and  Bff,  all  of  which  remain 
unchanged  in  the  successive  trials  with  assumed  values  of  J.  Thus 
we  obtain 

G  =  201°  T  57".4,        log  B  =  9.925092,        b  =  -f  0.568719, 
log  g  =  0.013500,  log  B"  =  9.510309,        b"  =  -f  0.959342. 

Then   we  assume,   by   means   of  approximate   elements   already 

known, 

log  J  =  0.397800, 

and  from 

V  cos  :'  cos  (jff  '  —  G)  =  J  cos  p  cos  (A  —  G)  +  g, 
V  cos  £'  sin  (H'  —G)  =  Acosp  sin  (A  —  G\ 
hr  sin  C'  =4  sin  p, 

we  find  U',  Cr,  and  A'.     These  give 
H'  =  153°  46'  20".5,         C'  =  +  7°  24'  16".4,         log  A'  =  0.487484. 


NUMERICAL   EXAMPLE.  215 

Next,  from 

cos  <p'  =  cos  C'  cos  /?"  cos  (A"  —  J2"')  -f-  sin  C'  sm/J", 
h'  cos  /  =  c,  h'  sin  ^'  =  (7, 

we  get 

log  C  =  9.912519,  c  =  4-  2.961673  ; 

and  from 


we  find 

log  r  =  0.323446. 
Then  we  have 


A"  =  c  =t  1/x2—  C2, 


from  which  to  find  J",  r",  and  x.     First,  by  means  of  the  approxi- 
mate elements,  we  assume 

log  J"  =  0.310000, 
which  gives  log  r"  =  0.053000,  and  hence  we  have 

TI  =  0.3783,  log  PL  =  0.002706,  log  x  =  0.090511. 

"With  this  value  of  x  we  obtain  from  the  expression  for  J",  the 
lower  sign  being  used,  since  J"  is  less  than  c, 

log  J"  =  0.309717. 

Repeating  the  calculation  of  r",  //,  and  x,  and  then  finding  J"  again,^ 

the  result  is 

log  J"  =  0.309647. 

Then,  by  means  of  the  formula  (67),  we  may  find  the  correct  value. 
Thus  we  have,  in  units  of  the  sixth  decimal  place, 

a  =  309717  —  310000  =  —  283,         a'  =  309647  —  309717  =  —  70, 
and  for  the  correction  to  the  last  result  for  log  J"  we  have 

___^_=_23. 

Therefore, 

log  J"  =  0.309624. 

By  means  of  this  value  we  get 

log  r"  =  0.052350,  log  *  =  0.090628, 


216  THEORETICAL   ASTRONOMY. 

and  tliis  value  of  JC  gives,  finally, 

log  A"  =  0.309623,  log  r"  =  0.052348. 

The  heliocentric  places  of  the  comet  are  now  found  from  the  equa- 
tions (71)  and  (72),  writing  A  cos/9  and  A"  cos  ft"  for  p  and  p", 
respectively.  Thus  we  obtain 

I  =  159°  43'  14".2,        b  =  +  10°  50'  14".0,        logr  =  0.323447, 
I"  =  144    17  47  .8,        b"  =  -f  35    14  28  .7,        log  r"  =  0.052347. 

The  agreement  of  these  results  for  r  and  r"  with  those  already 
obtained,  proves  the  accuracy  of  the  calculation.     Since  the  helio- 
centric longitudes  are  diminishing,  the  motion  is  retrograde. 
Then  from  (74)  we  get 

£  =  165°  17'  30".3,  t  =  63"  6'  32".5; 

and  from 


COS  I  COS  I 

we  obtain 

u  =  12°  10'  12".6,  u"  =  40°  18'  51".2, 

the  values  of  —  u  and  I  —  &  being  in  the  same  quadrant  when  the 
motion  is  retrograde.     The   equation   (79)  gives  log  x  =  0.090630, 
which  agrees  with  the  value  already  found. 
The  formulae  (81)  give 

<o  =  129°  6'  46".3,  log  q  =  9.760326, 

and  hence  we  have 
v  =  u  —  <*>  =  —  116°  56'  33".7,          o"  =  u"  —  a>  =  —  88°  47'  55".l, 

from  which  we  get 

T=  1858  Sept.  29.4274. 

From  these  elements  we  find 
log  r'  ==  0.212844,          vf  =  — 107°  7'  34".0,          u'  =  21°  59'  12".3, 

and  from 

tan  (f  —  &)  —  —  cos  i  tan  u', 

tan  bf  =  —  tan  i  sin  (J!  —  &  ), 
we  get 

I'  =  154°  56'  33".4,  6'  =  +  19°  30'  22".l. 


NUMERICAL   EXAMPLE.  217 

By  means  of  these  and  the  values  of  O'  and  Rf,  we  obtain 
A'  =  137°  39'  13".3,  p  =  +  12°  54'  45".3, 

and  comparing  these  results  with  observation,  we  have,  for  the  error 
of  the  middle  place, 

C.  — O. 

/  =  —  27".2,  A/5'  =  —  23".7. 


From  the  relative  positions  of  the  sun,  earth,  and  cctnefc  at  the 
time  t"  it  is  easily  seen  that,  in  order  to  diminish  these  residuals,  the 
geocentric  distance  must  be  increased,  and  therefore  we  assume,  for 
a  second  value  of  J, 

log  J  =  0.398500, 
from  which  we  derive 

H'  =  153°  44'  57".6,  C'  =  -f  7°  24'  26".l,      log  h'  =•  0.488026, 

log  C=  9.912587,  logc  =  0.472115,  logr  =  0.324207, 

log  A"  =  0.311054,  log  r"  =  0.054824,  log  x  =  0.089922, 

Then  we  find  the  heliocentric  places 

I  =  159°  40'  33".8,        b  =  +  10°  50'    8".6,        logr  =  0.324207, 
I"  =  144    17  12  .1,        b"  =  +  35     8  37  .8,        log  r"  =  0.054825, 

and  from  these, 

£  =  165°  15'  41".l,  i  =  63°    2'  49".2, 

u  =    12    10  30.8,  w"=:40    13  26.0, 

<o  =  128    54  44  .4,  log  q  =  9.763620, 

T=  1858  Sept.  29.8245,  log /  =  0.214116, 

v'  =  —  106°  55'  43".8,  u'  =      21°  59'    0".6, 

r=      154    5332.3,  V  =  +  19    2931.9, 

X=      137    3939.7,  /5'=  +  12    55     2.9. 

Therefore,  for  the  second  assumed  value  of  J,  we  have 

C.  — O. 

cos  jt  AA'  =  —  1".5,  A/5'  =  —  6".l. 

Since  these  residuals  are  very  small,  it  will  not  be  necessary  to 
make  a  third  assumption  in  regard  to  J,  but  we  may  at  once  derive 
the  correction  to  be  applied  to  the  last  assumed  value  by  means  of 
the  equations  (109).  Thus  we  have 

a'  =  —  1.5,        a  '  =  —  27.2,        d'  =  —  6.1,        d"  =  —  23.7, 
3  log  J  =  —  0.000700, 


218  THEORETICAL   ASTRONOMY. 

and,  expressing  d  log  A  in  units  of  the  sixth  decimal  place,  these 
equations  give 

25.7*  —  1050  =  0. 
17.&B  —  4270  =  0. 

Combining  these  according  to  the  method  of  least  squares,  we  get 

_  105X2.57 +  427X1.76 

(2.57)2  +  (1.76)2 

Hence  the  corrected  value  of  log  A  is 

log  A  =  0.398500  +  0.000106  =  0.398606. 

With  this  value  of  log  A  the  final  elements  are  computed  as  already 
illustrated,  and  the  following  system  is  obtained : — 

T=  1858  Sept.  29.88617  Washington  mean  time. 
TT  =   36°  22'  36".9  ] 
ft  =  165    15  24  .8  /   Mean  E(lumox  1858-°- 

t=   63      2  14.2 
log  ^  =  9.764142 

Motion  Retrograde. 

If  the  distinction  of  retrograde  motion  is  not  adopted,  and  we  regard 
i  as  susceptible  of  any  value  from  0°  to  180°,  we  shall  have 

7T  =  294°    8'12".7, 
*  =  116    57  45  .8, 

the  other  elements  remaining  the  same. 

The  comparison  of  the  middle  place  with  these  final  elements 
gives  the  following  residuals : — 

C.-O. 

cos  /5  AA  =  +  0".2,  A£  =  —  4".3. 

These  errors  are  so  small  that  the  orbit  indicated  by  the  observed 
places  on  which  the  elements  are  based  differs  very  little  from  a 
parabola. 

When,  instead  of  a  single  place,  a  series  of  intermediate  places  is 
employed  to  correct  the  assumed  value  of  J,  it  is  best  to  adopt  the 
equator  as  the  fundamental  plane,  since  an  error  in  a  or  d  will  affect 
both  A  and  /9;  and,  besides,  incomplete  observations  may  also  be  used 


NUMERICAL   EXAMPLE.  219 

when  the  fundamental  plane  is  that  to  which  the  observations  are 
directly  referred.  Further,  the  entire  group  of  equations  of  con- 
dition for  the  determination  of  x,  according  to  the  formulae  (109), 
must  be  combined  by  multiplying  each  equation  by  the  coefficient  of 
x  in  that  equation  and  taking  the  sum  of  all  the  equations  thus 
formed  as  the  final  equation  from  which  to  find  x,  the  observations 
being  supposed  equally  good. 


THEORETICAL   ASTRONOMY. 


CHAPTER  IV. 

DETERMINATION,  FROM  THREE  COMPLETE  OBSERVATIONS,  OF  THE  ELEMENTS  OP 
THE  ORBIT  OF  A  HEAVENLY  BODY,  INCLUDING  THE  ECCENTRICITY  OR  FORM  OB 
THE  CONIC  SECTION. 

77.  THE  formulae  which  have  thus  far  been  derived  for  the  deter- 
mination of  the  elements  of  the  orbit  of  a  heavenly  body  by  means 
of  observed  places,  do  not  suffice,  in  the  form  in  which  they  have 
been  given,  to  determine  an  orbit  entirely  unknown,  except  in  the 
particular  case  of  parabolic  motion,  for  which  one  of  the  elements 
becomes  known.  In  the  general  case,  it  is  necessary  to  derive  at 
least  one  of  the  curtate  distances  without  making  any  assumption  as 
to  the  form  of  the  orbit,  after  which  the  others  may  be  found.  But, 
preliminary  to  a  complete  investigation  of  the  elements  of  an  un- 
known orbit  by  means  of  three  complete  observations  of  the  body, 
it  is  necessary  to  provide  for  the  corrections  due  to  parallax  and  aber- 
ration, so  that  they  may  be  applied  in  as  advantageous  a  manner  as 
possible. 

When  the  elements  are  entirely  unknown,  we  cannot  correct  the 
observed  places  directly  for  parallax  and  aberration,  since  both  of 
these  corrections  require  a  knowledge  of  the  distance  of  the  body 
from  the  earth.  But  in  the  case  of  the  aberration  we  may  either 
correct  the  time  of  observation  for  the  time  in  which  the  light  from 
the  body  reaches  the  earth,  or  we  may  consider  the  observed  place 
corrected  for  the  actual  aberration  due  to  the  combined  motion  of  the 
earth  and  of  light  as  the  true  place  at  the  instant  when  the  light  left 
the  planet  or  comet,  but  as  seen  from  the  place  which  the  earth  occu- 
pies at  the  time  of  the  observation.  When  the  distance  is  unknown, 
the  latter  method  must  evidently  be  adopted,  according  to  which  we 
apply  to  the  observed  apparent  longitude  and  latitude  the  actual 
aberration  of  the  fixed  stars,  and  regard  this  place  as  corresponding 
to  the  time  of  observation  corrected  for  the  time  of  aberration,  to  be 
eifected  when  the  distances  shall  have  been  found,  but  using  for  the 
place  of  the  earth  that  corresponding  to  the  time  of  observation.  It- 
will  appear,  therefore,  that  only  that  part  of  the  calculation  of  the 


DETERMINATION   OF   AN   ORBIT.  221 

elements  which  involves  the  times  of  observation  will  have  to  be  re- 
peated after  the  corresponding  distances  of  the  body  from  the  earth 
have  been  found.  First,  then,  by  means  of  the  apparent  obliquity  of 
the  ecliptic,  the  observed  apparent  right  ascension  and  declination 
must  be  converted  into  apparent  longitude  and  latitude.  Let  ^0  and 
/?0,  respectively,  denote  the  observed  apparent  longitude  and  latitude; 
and  let  O0  be  the  true  longitude  of  the  sun,  2Q  its  latitude,  and  Rn 
its  distance  from  the  earth,  corresponding  to  the  time  of  observation. 
Then,  if  X  and  /9  denote  the  longitude  and  latitude  of  the  planet  or 
comet  corrected  for  the  actual  aberration  of  the  fixed  stars,  we  shall 
have 

A  —  J0  =  +  20".445  cos  (A  —  Q0)  sec/5  +  0".343  cos  (/I  —  281°)  sec ,9, .. 
£  —  /?„  =  —  20".445  sin  (A  —  Q0)  sin  ft  —  0".343  sin  (I  —  281°)  sin  /?. l  ; 

In  computing  the  numerical  values  of  these  corrections,  it  will  be 
sufficiently  accurate  to  use  ^0  and  /90  instead  of  X  and  /5  in  the  second 
members  of  these  equations,  and  the  last  terms  may,  in  most  cases, 
be  neglected.  The  values  of  X  and  /9  thus  derived  give  the  true  place 
of  the  body  at  the  time  t —  497*.78  J,  but  as  seen  from  the  place  of 
the  earth  at  the  time  t. 

When  the  distance  of  the  planet  or  comet  is'unknown,  it  is  impos- 
sible to  reduce  the  observed  place  to  the  centre  of  the  earth ;  but  if 
we  conceive  a  line  to  be  drawn  from  the  body  through  the  true  place 
of  observation,  it  is  evident  that  were  an  observer  at  the  point  of 
intersection  of  this  line  with  the  plane  of  the  ecliptic,  or  at  any  point 
in  the  line,  the  body  would  be  seen  in  the  same  direction  as  from  the 
actual  place  of  observation.  Hence,  instead  of  applying  any  correc- 
tion for  parallax  directly  to  the  observed  apparent  place,  we  may 
conceive  the  place  of  the  observer  to  be  changed  from  the  actual  place 
to  this  point  of  intersection  with  the  ecliptic,  and,  therefore,  it  be- 
comes necessary  to  determine  the  position  of  this  point  by  means  of 
the  data  furnished  by  observation. 

Let  00  be  the  sidereal  time  corresponding  to  the  time  t0  of  obser- 
vation, <p'  the  geocentric  latitude  of  the  place  of  observation,  and  p0 
the  radius  of  the  earth  at  the  place  of  observation,  expressed  in  parts 
of  the  equatorial  radius  as  unity.  Then  #0  is  the  right  ascension  and 
<pr  the  declination  of  the  zenith  at  the  time  t0.  Let  10  and  b0  denote 
these  quantities  converted  into  longitude  and  latitude,  or  the  longitude 
and  latitude  of  the  geocentric  zenith  at  the  time  t0.  The  rectangular 
co-ordinates  of  the  place  of  observation  referred  to  the  centre  of  the 


222  THEORETICAL   ASTRONOMY. 

earth  and  expressed  in  parts  of  the  mean  distance  of  the  earth  from 
the  sun  as  the  unit,  will  be 

x0  =  p9  sin  TTO  cos  b9  cos  19, 
y0  =  pQ  sin  rr0  cos  bQ  sin  I9f 
zQ  =  pQ  sin  TTO  sin  b0) 

in  which  TTO=  8".57116. 

Let  J0  be  the  distance  of  the  planet  or  comet  from  the  true  place 
of  the  observer,  and  J,  its  distance  from  the  point  in  the  ecliptic  to 
which  the  observation  is  to  be  reduced.  Then  will  the  co-ordinates 
of  the  place  of  observation,  referred  to  this  point  in  the  ecliptic,  be 

Xf  =  ( J,  —  J0)  COS  ft  COS  A, 

y,  =  (A,  —  4>)  cos  £  sin  ^ 
z,  =(4  —  4>)sin/?, 

the  axis  of  x  being  directed  to  the  vernal  equinox.  Let  us  now 
designate  by  O  the  longitude  of  the  sun  as  seen  from  the  point  of 
reference  in  the  ecliptic,  and  by  R  its  distance  from  this  point.  Then 
will  the  heliocentric  co-ordinates  of  this  point  be 

X=  —R  cos  0, 

t  T/" T>  Q^  s~\ 


The  heliocentric  co-ordinates  of  the  centre  of  the  earth  are 

XQ  =  —  RQ  cos  £0  cos  Q  0, 
YQ  =  —  R0  cos  T0  sin  O  0, 
Z0  =  —  R0  sin  J0. 

But  the  heliocentric  co-ordinates  of  the  true  place  of  observation 

will  be 

X+x,,  Y+yn  Z  +  z,, 

or 


and,  consequently,  we  shall  have 

R  cos  O  —  (4,  —  4>)  cos  ^  cos  A  =  /?0  cos  ro  cos  Q0  —  p0  sin  TTO  cos  50  cos  /0, 
R  sin  O  —  (4  —  ^0)  cos  £  sin  A  =  .R0  cos  ^0  sin  O0  —  Po  sin  ^o  cos  ^o  sm  ^o> 
—  (  J,  —  J0)  sin  /?         =  .Ro  sin  £0  —  ^?0  sin  ^0  sin  b0. 

If  we  suppose  the  axis  of  x  to  be  directed  to  the  point  whose  longi- 
tude is  O0>  these  become 


DETERMINATION   OF   AN   OKBIT.  223 

R  cos  (0  —  O0)  —  (4  —  4>)  cos  13  cos  (A  —  O0)  = 

EQ  cos  2"0  —  p0  sin  TTO  cos  60  cos  (70  —  O0), 
.#  sin  (O  —  Go)  —  (4  —  4.)  cos  /?  sin  (A  —  O0)  =  (2) 

—  p0  sin  TTO  cos  60  sin  (10  —  O0)> 
—  (  A,  —  J0)  sin  /?  —  RQ  sin  2"0  —  pQ  sin  TTO  sin  60, 

from  which  R  and  O  may  be  determined.     Let  us  now  put 

Z>;  (3) 


then,  since  ;r0,  Sw  and   O  —  O0  are  small,  these  equations  may  be 
reduced  to 

R  =  D  cos  (X  —  O0)  —  ^o  PQ  cos  fy>  cos  (10  —  Q0)  +  R0, 
R  (O  —  O0)  =  D  sin  (A  —  Q0)  —  7r0/?0  cos  b0  sin  (J0  —  Q8), 
0  —  Dtan/?  —  7r0^o0sin60  +  ^0^o- 

Hence  we  shall  have,  if  TTO  and  2"0  are  expressed  in  seconds  of  arc, 


(4) 


000.o 
206264.8 


,   206264.8  D  sin  (A  —  Q0)  —  TTO  /QO  cos  b0  sin  (^—  00) 

W  —  W0  ~T~  "  p  » 


from  which  we  may  derive  the  values  of  O  and  R  which  are  to  be 
used  throughout  the  calculation  of  the  elements  as  the  longitude  and 
distance  of  the  sun,  instead  of  the  corresponding  places  referred  to 
the  centre  of  the  earth.  The  point  of  reference  being  in  the  plane 
of  the  ecliptic,  the  latitude  of  the  sun  as  seen  from  this  point  is  zero, 
which  simplifies  some  of  the  equations  of  the  problem,  since,  if  the 
observations  had  been  reduced  to  the  centre  of  the  earth,  the  sun's 
latitude  would  be  retained. 

We  may  remark  that  the  body  would  not  be  seen,  at  the  instant 
of  observation,  from  the  point  of  reference  in  the  direction  actually 
observed,  but  at  a  time  different  from  tQ,  to  be  determined  by  the 
interval  which  is  required  for  the  light  to  pass  over  the  distance 
A,  —  J0.  Consequently  we  ought  to  add  to  the  time  of  observation 

the  quantity 

(  J,  —  J0)  497'.78  =  497'.7S  D  sec  /9,  (5; 

which  is  called  the  reduction  of  the  time  ;  but  unless  the  latitude  of 
the  body  should  be  very  small,  this  correction  will  be  insensible. 
The  value  of  A  derived  from  equations  (1)  and  the  longitude  O 


224  THEORETICAL   ASTRONOMY. 

derived  from  (4)  should  be  reduced  by  applying  the  correction  for 
nutation  to  the  mean  equinox  of  the  date,  and  then  both  these  and 
the  latitude  /9  should  be  reduced  by  applying  the  correction  for  pre- 
cession to  the  ecliptic  and  mean  equinox  of  a  fixed  epoch,  for  which 
the  beginning  of  the  year  is  usually  chosen. 

In  this  way  each  observed  apparent  longitude  and  latitude  is  to  be 
corrected  for  the  aberration  of  the  fixed  stars,  and  the  corresponding 
places  of  the  sun,  referred  to  the  point  in  which  the  line  drawn  from 
the  body  through  the  place  of  observation  on  the  earth's  surface  in- 
tersects the  plane  of  the  ecliptic,  are  derived  from  the  equations  (4). 
Then  the  places  of  the  sun  and  of  the  planet  or  comet  are  reduced 
to  the  ecliptic  and  mean  equinox  of  a  fixed  date,  and  the  results  thus 
obtained,  together  with  the  times  of  observation,  furnish  the  data  for 
the  determination  of  the  elements  of  the  orbit. 

When  the  distance  of  the  body  corresponding  to  each  of  the 
observations  shall  have  been  determined,  the  times  of  observation 
may  be  corrected  for  the  time  of  aberration.  This  correction  is 
necessary,  since  the  adopted  places  of  the  body  are  the  true  places 
for  the  instant  when  the  light  was  emitted,  corresponding  respectively 
to  the  times  of  observation  diminished  by  the  time  of  aberration, 
but  as  seen  from  the  places  of  the  earth  at  the  actual  times  of 
observation,  respectively. 

When  /?  =  0,  the  equations  (4)  cannot  be  applied,  and  when  the 
latitude  is  so  small  that  the  reduction  of  the  time  and  the  correction 
to  be  applied  to  the  place  of  the  sun  are  of  considerable  magnitude, 
it  will  be  advisable,  if  more  suitable  observations  are  not  available, 
to  neglect  the  correction  for  parallax  and  derive  the  elements,  using 
the  uncorrected  places.  The  distances  of  the  body  from  the  earth 
which  may  then  be  derived,  will  enable  us  to  apply  the  correction  for 
parallax  directly  to  the  observed  places  of  the  body. 

When  the  approximate  distances  of  the  body  from  the  earth  are 
already  known,  and  it  is  required  to  derive  new  elements  of  the 
orbit  from  given  observed  places  or  from  normal  places  derived  from 
many  observations,  the  observations  may  be  corrected  directly  for 
parallax,  and  the  times  corrected  for  the  time  of  aberration.  We 
shall  then  have  the  true  places  of  the  body  as  seen  from  the  centre 
of  the  earth,  and  if  these  places  are  adopted,  it  will  be  necessary,  for 
the  most  accurate  solution  possible,  to  retain  the  latitude  of  the  sun 
in  the  formulae  which  may  be  required.  But  since  some  of  these 
formulae  acquire  greater  simplicity  when  the  sun's  latitude  is  not 
introduced,  if,  in  this  case,  we  reduce  the  geocentric  places  to  the 


DETERMINATION   OF   AN   ORBIT.  225 

point  in  which  a  perpendicular  let  fall  from  the  centre  of  the  earth 
to  the  plane  of  the  ecliptic  cuts  that  plane,  the  longitude  of  the  sun 
will  remain  unchanged,  the  latitude  will  be  zero,  and  the  distance  R 
will  also  be  unchanged,  since  the  greatest  geocentric  latitude  of  the 
sun  does  not  exceed  1".  Then  the  longitude  of  the  planet  or  comet 
as  seen  from  this  point  in  the  ecliptic  will  be  the  same  as  seen  from 
the  centre  of  the  earth,  and  if  J,  is  the  distance  of  the  body  from 
this  point  of  reference,  and  /9,  its  latitude  as  seen  from  this  point,  we 
shall  have 

A,  cos  /?,  =  A  cos  /9, 

At  sin  ft,  =  A  sin  /5  —  J?0  sin  S0, 

from  which  we  easily  derive  the  correction  /?,  —  /9,  or  A/9,  to  be  applied 
to  the  geocentric  latitude.  Thus,  we  find 

A/9  =  -^>  cos /9,  (6) 

~t 

2Q  being  expressed  in  seconds.  This  correction  having  been  applied 
to  the  geocentric  latitude,  the  latitude  of  the  sun  becomes 

2=0. 

The  correction  to  be  applied  to  the  time  of  observation  (already 
diminished  by  the  time  of  aberration)  due  to  the  distance  J,  —  J0 
will  be  absolutely  insensible,  its  maximum  value  not  exceeding 
Os.002.  It  should  be  remarked  also  that  before  applying  the  equa- 
tion (6),  the  latitude  JT0  should  be  reduced  to  the  fixed  ecliptic  which 
it  is  desired  to  adopt  for  the  definition  of  the  elements  which  deter- 
mine the  position  of  the  plane  of  the  orbit. 

78.  When  these  preliminary  corrections  have  been  applied  to  the 
data,  we  are  prepared  to  proceed  with  the  calculation  of  the  elements 
of  the  orbit,  the  necessary  formulae  for  which  we  shall  now  investi- 
gate. For  this  purpose,  let  us  resume  the  equations  (6)3 ;  and,  if  we 
multiply  the  first  of  these  equations  by  tan /9  sin  A" —  tan /9"  sin  A, 
the  second  by  tan/3"  cos  A  —  tan  /9  cos  A",  and  the  third  by  sin  (X  —  )"\ 
and  add  the  products,  we  shall  have 

0  —  nR  (tan  /9"  sin  (A  —  Q)  —  tan  /9  sin  (A"  —  Q)) 

—  p'  (tan  /?  sin  (A"  —  A')  —  tan  jf  sin  (A"  —  A)  -f-  tan  $'  sin  (A'  —  A)) 

—  R'  (tan  P"  sin  (A  —  Q')  —  tan  ft  sin  (A"  —  ©')) 

+  n"R"  (tan  ft"  sin  (A  —  Q")  —  tan  /5  sin  (A"  —  0")). 

It  should  be  observed  that  when  the  correction  for  parallax  is  applied 

15 


226  THEORETICAL   ASTRONOMY. 

to  the  place  of  the  sun,  pf  is  the  projection,  on  the  plane  of  the 
ecliptic,  of  the  distance  of  the  body  from  the  point  of  reference  to 
which  the  observation  has  been  reduced. 

Let  us  now  designate  by  jfiTthe  longitude  of  the  ascending  node, 
and  by  I  the  inclination  to  the  ecliptic,  of  a  great  circle  passing 
through  the  first  and  third  observed  places  of  the  body,  and  we  have 

tan  p  —  sin  (A  —  K}  tan  J, 
tan  0"  =  sin  (A"  —  JE")tanJ.. 

Introducing  these  values  of  tan  0  and  tan  ft"  into  the  equation  (7), 
since 

sin  (A  —  O)  sin  (A"  —  K)  —  sin  (A"  —  Q)  sin  (A  —  JT)  = 

—  sin  (A"  —  A)  sin  (Q  —  JT), 
sin  (A'  —  A)  sin  (A"  —  JT)  +  sin  (A"  —  A')  sin  (A  —  JT)  = 

-f-  sin  (A"  —  A)  sin  (A'  —  K\ 
sin  (A  —  O')  sin  (A"  —  K)  —  sin  (A"  —  G')  sin  (A  —  JT)  = 

—  sin  (A"  —  A)  sin  (G'  —  JD, 
sin  (A  —  0")  sin  (A"  —  K)  —  sin  (A"  —  0")  sin  (A  —  K)  = 

—  sin  (A"  —  A)  sin  (G"  —  K\ 

we  obtain,  by  dividing  through  by  sin  (A"  —  A)  tan  /, 

0  =  nR  sin  (G  —  -ET)  +  p'  (sin  (A'  —  K}  —  tan  pf  cot  I) 

Let  /90  denote  the  latitude  of  that  point  of  the  great  circle  passing 
through  the  first  and  third  places  which  corresponds  to  the  longitude 
^',  then 

tan  ft  =  sin  (A'  —  K)  tan  J, 

and  the  coefficient  of  pr  in  equation  (9)  becomes 

sin  (ft  —  ff') 
cos  ft  cos  pr  tan  / 
Therefore,  if  we  put 


we  shall  have 


Bn       - 
"»=  cos  /».  tan/' 


,, 


This  formula  will  give  the  value  of  pft  or  of  J',  when  the  values  of 
n  and  n"  have  been  determined,  since  a0  and  K  are  derived  from  the 
data  furnished  by  observation. 


DETERMINATION   OF   AN   OEBIT.  227 

To  find  K  and  7,  we  obtain  from  equations  (8)  by  a  transformation 
precisely  similar  to  that  by  which  the  equations  (75)3  were  derived, 


We  may  also  compute  K  and  I  from  the  equations  which  may  be 
derived  from  (74)3  and  (76)3  by  making  the  necessary  changes  in  the 
notation,  and  using  only  the  upper  sign,  since  I  is  to  be  taken  always 
less  than  90°. 

Before  proceeding  further  with  the  discussion  of  equation  (11),  let 
us  derive  expressions  for  p  and  p"  in  terms  of  ,o',  the  signification  of 
p  and  ptfy  when  the  corrections  for  parallax  are  applied  to  the  places 
of  the  sun,  being  as  already  noticed  in  the  case  of  pf. 

79.  If  we  multiply  the  first  of  equations  (6)3  by  sin  0"  tan/3", 
the  second  by  —  cos  0"  tan/3",  and  the  third  by  sin  (A"  —  ©")>  and 
add  the  products,  we  get 

0=w/>(tan£"sin(0"—  A)—  tan/9sin(0"—  *"))—  njRtan/9"8in(0"—  0) 
-V  (tan  p'  sin  (0"—  A')—  tan  f  sin  (©"—  A"))+#  tan  p  sin  (©"—  Q'), 

(13) 
which  may  be  written 

0=  np  (tan  /?  sin  (A"—  0")—  tan  /?"  sin  (X—  ©  "))—  rc^tan  0"  sin  (0"—  0) 
-f  ft  (tan  /?"  sin  (A'  —  Q  ")  —  tan  ft  sin  (A"  —  ©  ")) 

-VCtan/S'  —  tanft)sin(A"  —  Q")  +  #  tan/9';  sin(Or;—  00- 

Introducing  into  this  the  values  of  tan  ft,  tan  /3",  and  tan  /30  in  terms 
of  1  and  jfif,  and  reducing,  the  result  is 

Q  =  np  sin  (A"—  A)  sin  (  0  "—  K)  —  nR  sin  (  ©  "  -  0  )  sin  (A"—  £") 


+  Rr  sin  (0"—  00  sin  W  —  -K"). 
Therefore  we  obtain 


_^/sin(^—  AQ  q0sec^        sin(^—  Q^)  \ 

=~  w  \  sin  (A"  —  A)  +  sin  (A"  —  A)  '  sin  (Q"  —  K}  / 


sn      — 


/i  sin  (A"  —  A)sin(O"—  JE") 

But,  by  means  of  the  equations  (9)3,  we  derive 

"—  0)  =  (N—  ri)  Rsm(O"-  0), 


228  THEORETICAL   ASTRONOMY. 

and  the  preceding  equation  reduces  to 

sin  (A"  —  0") 


n\sin(A"  — A)    '   sin(A"— A)"sin(©"— _y/  (u. 

N\Ksm(Q"—  Q)8m(A"  —  K) 
"~rT/     sin  (A"  —  A)sin(O"  —  K)    ' 

To  obtain  an  expression  for  prf  in  terms  of  pr,  if  we  multiply  the 
first  of  equations  (6)3  by  sin  ©  tan/9,  the  second  by  — cos  ©  tan/9, 
and  the  third  by  sin  (X  —  ©),  and  add  the  products,  we  shall  have 

0=»V'(tan^sin(A'/— ©)—  tan /S" sin  (A— ©))— n"K"tan£sin(©"— ©) 
*'— ©)— tan/?' sin  (A— ©))+JR'tan/5sin(©'— ©).  (15) 


Introducing  the  values  of  tan  /9,  tan  /9r,  and  tan  /9r/  in  terms  of  K  and 
/,  and  reducing  precisely  as  in  the  case  of  the  formula  already  found 
for  p,  we  obtain 

„ p'  I  sin  (A'  —  A)  a0  sec  j?        sin  (A  —  ©  )  \ 

n"  \  sin  (A"  —  A)       sin  (A"  —  A)    sin  (©  —  K)  / 


T^et  us  now  put,  for  brevity, 

.       Esm(Q-K)  _R'sm(Qf—K) 

o  =  -  —  j  c  —  i 

a0  a0 

^_^sin(0//—  g)  sec/gf  ,  _.Rfi"8in(0"—  0) 

a  J  ""  sin  (A"  —  A)' 


in(^—  JQ  ,       A  sin  (A  —  .g) 

- 


and  the  equations  (11),  (14),  and  (16)  become 

(18) 


If  n  and  n"  are  known,  these  equations  will,  in  most  cases,  be 
sufficient  to  determine  p,  //,  and  //'. 


DETEKMINATION   OF   AN   OKBIT.  229 

80.  It  will  be  apparent,  from  a  consideration  of  the  equations 
which  have  been  derived  for  p,  pf,  and  plft  that  under  certain  circum- 
stances they  are  inapplicable  in  the  form  in  which  they  have  been 
given,  and  that  in  some  cases  they  become  indeterminate.  When  the 
great  circle  passing  through  the  first  and  third  observed  places  of  the 
body  passes  also  through  the  second  plaoe,  we  have  a0  =  0,  and 
equation  (11)  reduces  to 

n"R"  sin (Q"  —  K)  +  nR  sin (0  —  K}  =  R'  sin (0'  —  K). 

If  the  ratio  of  n"  to  n  is  known,  this  equation  will  determine  the 
quantities  themselves,  and  from  these  the  radius-vector  rr  for  the 
middle  place  may  be  found.  But  if  the  great  circle  which  thus 
passes  through  the  three  observed  places  passes  also  through  the 
second  place  of  the  sun,  we  shall  have  K==  O',  or  K=  180°  -j-  O', 
and  hence 

n"R"  sin  (O"  —  0')  —  nR  sin  (©'  —  O)  =  0, 
or 

n"        Jg8in(0'  —  0) 
n  ~.R"sm(O"—  O')' 

from  which  it  appears  that  the  solution  of  the  problem  is  in  this 
case  impossible. 

If  the  first  and  third  observed  places  coincide,  we  have  ^  =  X"  and 
/9  =  /3",  and  each  term  of  equation  (7)  reduces  to  zero,  so  that  the 
problem  becomes  absolutely  indeterminate.  Consequently,  if  the 
data  are  nearly  such  as  to  render  the  solution  impossible,  according 
to  the  conditions  of  these  two  cases  of  indetermination,  the  elements 
which  may  be  derived  will  be  greatly  affected  by  errors  of  observa- 
tion. If,  however,  A  is  equal  to  A"  and  /9"  differs  from  /9,  it  will  be 
possible  to  derive  pf,  and  hence  p  and  p";  but  the  formulae  which 
have  been  given  require  some  modification  in  this  particular  case. 
Thus,  when  J  =  A",  we  have  K=X'  =  19  7=90°,  and  ft  =90°, 

and  hence  a0,  as  determined  by  equation  (10),  becomes  -     Still,  in 

this  case  it  is  not  indeterminate,  since,  by  recurring  to  the  original 
equation  (9),  the  coefficient  of  pf,  which  is  —  a0  sec  ft',  gives 

a0  =  sin  p  cot  /—  cos  /?'  sin  (/  —  K)t  (19) 

and  when  A  =  A",  it  becomes  simply 

«0  =  —  cos  p  sin  (A'  —  K). 


— 

/          N"\ 
\          n"  / 


230  THEORETICAL   ASTRONOMY. 

Whenever,  therefore,  the  difference  A"  —  A  is  very  small  compared 
with  the  motion  in  latitude,  a0  should  be  computed  by  means  of  the 
equation  (19)  or  by  means  of  the  expression  which  is  obtained 
directly  from  the  coefficient  of  p1  in  equation  (7). 

When  A  =  X"  =  K,  the  values  of  Jfw  Jtf/',  M2,  and  M2"  cannot 
be  found  by  means  of  the  equations  (17);  but  if  we  use  the  original 
form  of  the  expressions  for  p  and  p"  in  terms  of  pf,  as  given  by 
equations  (13)  and  (15),  without  introducing  the  auxiliary  angles, 
we  shall  have 

=  ft_   tan  p  sin  (A"  —  Q")  —  tan  p'  sin  (A'  —  Q") 
n  '  tan  ft  sin  (A"  —  0")  —  tan  p'  sin  (A  —  0") 

/     __N_\  _  jRtan£"sin(0"—  Q)  _ 
M  n   I  tan  ft  sin  (/I"  —  0")  —  tan  ft"  sin  (A  -  0")1 

tan  /?  sin  (A'  —  Q)  —  tan  p  sin  (A  —  Q) 
tan  /9  sin  (A"  —  0)  —  tan  p"  sin  (A  —  0) 

jR"tan£sin(0"--  0) 
tan  /?  sin  (A"  —  0)  —  tan  p'  sin  (A  —  ©)' 
Hence 

_  tan  /?'  sin  (X*  —  ©")  —  tan  p'  sin  (A'  —  Q") 
1  ~~  tan  0  sin  (A"  —  0")  —  tan  ft"  sin  (A  —  0")  ' 
„_       tan  ft  sin  (A'  —  0)  —  tan  /?'  sin  (A  —  Q) 
MI   Z  '  tan  /S  sin  (A"  -  0)  -  tan  0"  sin  (*—©)'  ,     , 

_  J?  tan  ^  sin  (0^—0)  _ 
8  ~  tan  /5  sin  (A"  —  0")  —  tan  ft"  sin  (A  —  ©")' 
„  _  _  R"  tan  ft  sin  (Q"—  0)  __ 
a   =  =  tan  ft  sin  (A"  —  0)  —  tan  ft"  sin  (A  —  ©)  ' 

are  the  expressions  for  Mlt  -Mi",  Mz,  and  M2"  which  must  be  used 
when  X  =  X'r  or  when  A  is  very  nearly  equal  to  A";  and  then  p  and  prr 
will  be  obtained  from  equations  (18).  These  expressions  will  also  be 
used  when  A"  —  X  =  180°,  this  being  an  analogous  case. 

When  the  great  circle  passing  through  the  first  and  third  observed 
places  of  the  body  also  passes  through  the  first  or  the  third  place  of 
the  sun,  the  last  two  of  the  equations  (18)  become  indeterminate,  and 
other  formula  must  be  derived.  If  we  multiply  the  second  of  equa- 
tions (7)3  by  tan/3"  and  the  fourth  by  —  sin  (A"—  Or),  and  add  the 
products,  then  multiply  the  second  of  these  equations  by  tan  /?  and 
the  fourth  by  —  sin  (A  —  ©'),  and  add,  and  finally  reduce  by  means 
of  the  relation 

NE  sin  (©'  -  0)  =  N"R"  sin  (0"  -  ©'), 
we  get 


o    —  — 


DETERMINATION   OF   AN   ORBIT.  231 

tan  p'  sin  (A'  —  ©')  —  tan  ft'  sin  (A"  —  ©') 


n    tan  ft"  sin  (A  —  ©')  —  tan  ft  sin  (A"  —  © ') 

.#"  tan/S"  sin  (©"—  0') 


tan  ft"  sin  (A  —  ©')  —  tan  ft  sin  (A"— 
tan  jt  sin  (A  —  0 ')  —  tan  ft  sin  (A'  —  0 ') 
tan  ft"  sin  (A  —  © ')  —  tan  ft  sin  (A"  —  ©') 

R  tan  /5  sin  (©'—©) 


(21) 


tan/S"  sin  (A  —  0')  —  tan/?  sin  (A"—  0') 
These  equations  are  convenient  for  determining  p  and  p'r  from  //  ; 
but  they  become  indeterminate  when  the  great  circle  passing  through 
the  extreme  places  of  the  body  also  passes  through  the  second  place 
of  the  sun.  Therefore  they  will  generally  be  inapplicable  for  the 
cases  in  which  the  equations  (18)  fail. 

If  we  eliminate  p"  from  the  first  and  second  of  the  equations  (6)3 
we  get 

Q  =  np  sin  (A"  —  A)  —  nR  sin  (A"  —  Q)  —  p'  sin  (A"  —  A') 

+  R'  sin  (A"  —  0')  —  ri'R"  sin  (A"  —  0"), 
from  which  we  derive 

p>    sin(A"-AQ 
-;Tsin(A"-A) 

nR  sin  (A"  —  Q  )  —  R  sin  (A"  —  Q')  +  ri'R'  sin  (I"  —  Q  ") 

n  sin  (A"  —  A) 
Eliminating  p  between  the  same  equations,  the  result  is 

P'    sin  (A'-  A) 
p    -n"    sin  (A"  -A) 

nR  sin  (A  —  Q)  —  R'  sin  (A  —  Q')  -f-  n"R"  sin  (A  —  Q") 

n"  sin  (A"  —  A) 

These  formulae  will  enable  us  to  determine  p  and  p"  from  p1  in  the 
special  cases  in  which  the  equations  (18)  and  (21)  are  inapplicable; 
but,  since  they  do  not  involve  the  third  of  equations  (6)3,  they  are 
not  so  well  adapted  to  a  complete  solution  of  the  problem  as  the 
formulae  previously  given  whenever  these  may  be  applied. 

If  we  eliminate  successively  p"  and  p  between  the  first  and  fourth 
of  the  equations  (7)3,  we  get 

tan  P"  cos  (X  —  Q  ')  —  tan  p  cos  (X1  —  Q  ') 


_ 
p     ~ 


„_ 
P     ~" 


n     tan  /3"  cos  (A  —  Q')  —  tan  p  cos  (A"  —  Q') 

tan/3"    nR  cos  (0'—  Q  )  —  R  +  n"R"  cos(Q"—  Q') 
n          tan  /5"  cos  (A  —  Q  ')  —  tan  p  cos  (A"  —  O')    ' 
tan  j3'  cos  (A  —  Q')  —  tan  /9  cos  (A'—  Q  ') 
n"'  tan  ft"  cos  (A  —  Q  ')  —  tan  ft  cos  (A"—  Q') 

tan/9    nR  cos  (Q'  —  Q)  —  JT  +  ri'R"  cos  (0"  —  0') 
n"    '       tan  0"  cos  (A—  ©')  —  tan/?  cos  (A"—  0')      ' 


232  THEORETICAL   ASTRONOMY. 

which  may  also  be  used  to  determine  p  and  pff  when  the  equations 
(18)  and  (21)  cannot  be  applied.  When  the  motion  in  latitude  is 
greater  than  in  longitude,  these  equations  are  to  be  preferred  instead 
of  (22)  and  (23.) 

81.  It  would  appear  at  first,  without  examining  the  quantities  in- 
volved in  the  formula  for  pf ,  that  the  equations  (26)3  will  enable  us 
<o  find  n  and  n"  by  successive  approximations,  assuming  first  tEat 

r  ..       T" 

n  =  r  n  =-,, 

and  from  the  resulting  value  of  p'  determining  rf,  and  then  carrying 
the  approximation  to  the  values  of  n  and  n"  one  step  farther,  so  as 
to  include  terms  of  the  second  order  with  reference  to  the  intervals 
of  time  between  the  observations.  But  if  we  consider  the  equation 
(10),  we  observe  that  a0  is  a  very  small  quantity  depending  on  the 
difference  ft'  —  /90,  and  therefore  on  the  deviation  of  the  observed 
path  of  the  body  from  the1  arc  of  a  great  circle,  and,  as  this  appears 
in  the  denominator  of  terms  containing  n  and  n"  in  the  equation 
(11),  it  becomes  necessary  to  determine  to  what  degree  of  approxi- 
mation these  quantities  must  be  known  in  order  that  the  resulting 
value  of  p'  may  not  be  greatly  in  error. 

To  determine  the  relation  of  a0  to  the  intervals  of  time  between 
the  observations,  we  have,  from  the  coefficient  of  p'  in  equation  (7), 

a0  sec  p  =  tan  ft  sin  (A"  —  A')  —  tan  /3'  sin  (A"  —  A)  +  tan/3"  sin  (A'  —  A). 
We  may  put 


tan/?  =tany3'  —  AT"  +  BT"*  —  ...., 
tan/5"  =  tan/5'  +  A?  +  Br*  +  ...., 

and  hence  we  have 

«0  sec  jf  =  (sin  (A"  —  A')  —  sin  (A"  —A)  +  sin  (A'  —  A))  tan  ? 
-f  (r  sin  (A'— A)  —  r"  sin  (A"— A'))  ^+(r2  sin  (A'— A)-f-r"2  sin  (A"—/))  B+. ., 

which  is  easily  transformed  into 

«0  sec  p  =  4  sin  J  (A'  —  A)  sin  \  (A"  —  A')  sin  j  (A"—  A)  tan  p      (25 ) 
-f  (r  sin  (A'— A)— T"  sin  (A"— A'))-A-f(r2  sin  (A'— A)+r"2  sin  (A"— X))B+. . . , 

If  we  suppose  the  intervals  to  be  small,  we  may  also  put 

sinJ-(A"  —  A)  =  i(A"  —  A), 
and 

sin  (A"  —  A)  =r.  A"  —  A,  sin  (A'  -  A)  =  A'  —  A. 


DETERMINATION   OF   AN   ORBIT.  233 

Further,  we  may  put 


Substituting  these  values  in  the  equation  (25),  neglecting  terms  of 
the  fourth  order  with  respect  to  r,  and  reducing,  we  get 

a0  =  TT'T"  (p's  tan  /?'  +  A'B  —  AB'}  cos  jt. 

It  appears,  therefore,  that  a0  is  at  least  of  the  third  order  with 
reference  to  the  intervals  of  time  between  the  observations,  and  that 
an  error  of  the  second  order  in  the  assumed  values  of  n  and  n"  may 
produce  an  error  of  the  order  zero  in  the  value  of  pr  as  derived  from 
equation  (11)  even  under  the  most  favorable  circumstances.  Hence, 
in  general,  we  cannot  adopt  the  values 


omitting  terms  of  the  second  order,  without  affecting  the  resulting 
value  of  pr  to  such  an  extent  that  it  cannot  be  regarded  even  as  an 
approximation  to  the  true  value  ;  and  terms  of  at  least  the  second 
order  must  be  included  in  the  first  assumed  values  of  n  and  nff. 
The  equation  (28)3  gives 


/"/  7* 

omitting  the  term  multiplied  by  -^~,  which  term  is  of  the  third  order 

/V) 

with  respect  to  the  times  ;  and  hence  in  this  value  of  —  ,  only  terms 

of  at  least  the  fourth  order  are  neglected.     Again,  from  the  equations 
(26)3  we  derive,  since  r'  —  r  -J-  r/r, 

n  +  n"=l  +  ~,  (27) 

in  which  only  terms  of  the  fourth  order  have  been  neglected.     Now 
the  first  of  equations  (18)  may  be  written  : 


p>  sec  /?'  =  (n  +  n")  -  n—  -  c,  (28) 


in  which,  if  we  introduce  the  values  of  —  and  n  +  nrr  as  given  by 

fi 

(26)  and  (27),  only  terms  of  the  fourth  order  with  respect  to  the 


234  THEORETICAL   ASTRONOMY. 

times  will  be  neglected,  and  consequently  the  resulting  value  of  />' 
will  be  affected  with  only  an  error  of  the  second  order  when  a0  is  of 
the  third  order.  Further,  if  the  intervals  between  the  observations 
are  not  very  unequal,  r2  —  r"2  will  be  a  quantity  of  an  order  superior 
to  r2,  and  when  these  intervals  are  equal,  we  have,  to  terms  of  the 
fourth  order, 

?L  =  !L' 

n          r' 

The  equation  (27)  gives 

2/8  (n  +  n"  —  1)  =  TT". 
Hence,  if  we  put 

P-n- 

n'  (29) 


we  may  adopt,  for  a  first  approximation  to  the  value  of  p', 


and  pf  will  be  affected  with  an  error  of  the  first  order  when  the  in- 
tervals are  unequal  ;  but  of  the  second  order  only  when  the  intervals 
are  equal.  It  is  evident,  therefore,  that,  in  the  selection  of  the 
observations  for  the  determination  of  an  unknown  orbit,  the  in- 
tervals should  be  as  nearly  equal  as  possible,  since  the  nearer  they 
approach  to  equality  the  nearer  the  truth  will  be  the  first  assumed 
values  of  P  and  §,  thus  facilitating  the  successive  approximations  ; 
and  when  a0  is  a  very  small  quantity,  the  equality  of  the  intervals 
is  of  the  greatest  importance. 
From  the  equations  (29)  we  get 


1 1/          _^  (  Tf-\  1       •*•      ~T          ;\        t-t       \\  /QIA 


and  introducing  P  and  Q  into  (28),  there  results 


This  equation  involves  both  pf  and  r'  as  unknown  quantities,  but 
by  means  of  another  equation  between  these  quantities  pf  may  be 
eliminated,  thus  giving  a  single  equation  from  which  rf  may  be 
found,  after  which  pf  may  also  be  determined. 


DETERMINATION   OF   AN   ORBIT.  235 

82.  Let  ty  represent  the  angle  at  the  earth  between  the  sun  and 
planet  or  comet  at  the  second  observation,  and  we  shall  have,  from 
the  equations  (93)3, 

tan/97 


tan  wr  =  — 


sin  (A'  —  Q') 

-,  (33^ 


cos  *'  =  cos  p  cos  (A'  —  0'), 

by  means  of  which  we  may  determine  ij/,  which  cannot  exceed  180°. 
Since  cos  ft'  is  always  positive,  cos  i//  and  cos  (A'  —  O  ')  must  have  the 
same  sign. 

We  also  have 

/2  =  j.»  +  jgr.  _  2  A'R  cos  4,', 


which  may  be  put  in  the  form 

r'2  =  0>'  sec  /3'  —  R  cos  4?  -f  R2  sin2  4', 
from  which  we  get 

P'  sec  ,3'  =  R'  cos  4'  ±  VV*—  jR"sin«4/.  (34) 

Substituting  for  /)r  sec  ^r  its  value  given  by  equation  (32),  we  have 


For  brevity,  let  us  put 

4,  —  c  =  km  (35) 

'     JcoS  =  4> 

and  we  shall  have 

k0  —  ^  =  R  cos  V  ±  1/r'2  — jR'2sm2V.  (36) 

When  the  values  of  P  and  Q  have  been  found,  this  equation  will 
give  the  value  of  r'  in  terms  of  quantities  derived  directly  from  the 
data  furnished  by  observation.  We  shall  now  represent  by  z'  the 
angle  at  the  planet  between  the  sun  and  earth  at  the  time  of  the 
second  observation,  and  we  shall  have 


sm  z' 


236  THEORETICAL   ASTRONOMY. 

Substituting  this  value  of  rf,  in  the  preceding  equation,  there  results 

(&0  -  E'  cos  40  sin  z'  +  K  sin  *'  cos  z'  =  j^jj^-,>  C38  ) 

and  if  we  put 

TJQ  sin  C  =  K  sin  V, 

,o  cos  Z  =  lcQ  —  R'  cos  *',  (39) 


the  condition  being  imposed  that  ra0  shall  always  be  positive,  we 

have,  finally, 

sin  (X  q=  C)  =  m0  sinV.  (40) 

In  order  that  m0  may  be  positive,  the  quadrant  in  which  £  is  taken 
must  be  such  that  %  shall  have  the  same  sign  as  10,  since  sin  i//  is 
always  positive. 

From  equation  (37)  it  appears  that  sin  z'  must  always  be  positive, 
or  2'  <  180°;  and  further,  in  the  plane  triangle  formed  by  joining 
the  actual  places  of  the  earth,  sun,  and  planet  or  comet  corresponding 
to  the  middle  observation,  we  have 

y      /  sin  (X  +  4/)       K  sin  (zr  +  *Q 


sin  4/  sin  z' 

Therefore, 


and,  since  pr  is  always  positive,  it  follows  that  sin  (z'  +  ty)  must  be 
positive,  or  that  z1  cannot  exceed  180°  —  ty. 

When  the  planet  or  comet  at  the  time  of  the  middle  observation  is 
both  in  the  node  and  in  opposition  or  conjunction  with  the  sun,  we 
shall  have  /?'  =  0,  '\J/  =  180°  when  the  body  is  in  opposition,  and 
•vJ/  =  0°  when  it  is  in  conjunction.  Consequently,  it  becomes  impos- 
sible to  determine  rr  by  means  of  the  angle  zr  ;  but  in  this  case  the 
equation  (36)  gives 

*.-£=-#+.', 

when  the  body  is  in  opposition,  the  lower  sign  being  excluded  by  the 
condition  that  the  value  of  the  first  member  of  the  equation  must  be 
positive,  and  for  \£/  =  0, 


the  upper  sign  being  used  when  the  sun  is  between  the  earth  and  the 


DETERMINATION    OF   AN   ORBIT.  237 

planet,  and  the  lower  sign  when  the  planet  is  between  the  earth  and 
the  sun.  It  is  hardly  necessary  to  remark  that  the  case  of  an  obser- 
vation at  the  superior  conjunction  when  /9'  =  0,  is  physically  impos- 
sible. The  value  of  r'  may  be  found  from  these  equations  by  trial ; 
and  then  we  shall  have 

when  the  body  is  in  opposition,  and 

/   T}f  fiJ 

when  it  is  in  inferior  conjunction  with  the  sun. 

For  the  case  in  which  the  great  circle  passing  through  the  extreme 
observed  places  of  the  body  passes  also  through  the  middle  place, 
which  gives  a0  =  0,  let  us  divide  equation  (32)  through  by  c,  and  we 
have 

*"      c       1  p'  sec  /?' 

~+7~  ~~T~' 

The  equations  (17)  give 


and  if  we  put 


c          c  _  r 

~1 ; 7T  ^>n- 


we  shall  have 

since  c  =  GO  when  a0  =  0.     Hence  we  derive 

(42) 

But  when  the  great  circle  passing  through  the  three  observed  places 
passes  also  through  the  second  place  of  the  sun,  both  c  and  C0  be- 
come indeterminate,  and  thus  the  solution  of  the  problem,  with  the 
given  data,  becomes  impossible. 

83.  The  equation  (40)  must  give  four  roots  corresponding  to  each 
sign,  respectively;  but  it  may  be  shown  that  of  these  eight  roots  at 
least  four  will,  in  every  case,  be  imaginary.  Thus,  the  equation  may 
be  written 

m0  sin4  z'  —  sin  z'  cos  C  —  +  cos  z'  sin  C, 


238  THEORETICAL   ASTRONOMY. 

and,  by  squaring  and  reducing,  this  becomes 

ra02  sin8  z'  —  2m0  cos  C  sin5  /  -f-  sm2  *'  —  sm2  C  =  0. 

When  £  is  within  the  limits  —  90°  and  +  90°,  cos  £  will  be  positive, 
and,  m0  being  always  positive,  it  appears  from  the  algebraic  signs  of 
the  terms  of  the  equation,  according  to  the  theory  of  equations,  that 
in  this  case  there  cannot  be  more  than  four  real  roots,  of  which  three 
will  be  positive  and  one  negative.  When  £  exceeds  the  limits  —  90° 
and  -f-  90°,  cos  £  will  be  negative,  and  hence,  in  this  case  also,  there 
cannot  be  more  than  four  real  roots,  of  which  one  will  be  positive 
and  three  negative.  Further,  since  sin2  £  is  real  and  positive,  there 
must  be  at  least  two  real  roots  —  one  positive  and  the  other  negative 
—  whether  cos  £  be  negative  or  positive. 

"We  may  also  remark  that,  in  finding  the  roots  of  the  equation  (40), 
it  will  only  be  necessary  to  solve  the  equation 

sin  (z  —  0  =  ra0  sin*  /,  (43) 

since  the  lower  sign  in  (40)  follows  directly  from  this  by  substituting 
180°  —  zr  in  place  of  2';  and  hence  the  roots  derived  from  this  will 
comprise  all  the  real  roots  belonging  to  the  general  form  of  the 
equation. 

The  observed  places  of  the  heavenly  body  only  give  the  direction 
in  space  of  right  lines  passing  through  the  places  of  the  earth  and 
the  corresponding  places  of  the  body,  and  any  three  points,  one  in 
each  of  these  lines,  which  are  situated  in  a  plane  passing  through  the 
centre  of  the  sun,  and  which  are  at  such  distances  as  to  fulfil  the 
condition  that  the  areal  velocity  shall  be  constant,  according  to  the 
relation  expressed  by  the  equation  (30)1?  must  satisfy  the  analytical 
conditions  of  the  problem.  It  is  evident  that  the  three  places  of  the 
earth  may  satisfy  these  conditions  ;  and  hence  there  may  be  one  root 
of  equation  (43)  which  will  correspond  to  the  orbit  of  the  earth,  or 

give 

,'  =  0. 

Further,  it  follows  from  the  equation  (37)  that  this  root  must  be 


and  such  would  be  strictly  the  case  if,  instead  of  the  assumed  values 
of  P  and  §,  their  exact  values  for  the  orbit  of  the  earth  were  adopted, 
and  if  the  observations  were  referred  directly  to  the  centre  of  the 
earth,  in  the  correction  for  parallax,  neglecting  also  the  perturbation* 
in  the  motion  of  the  earth. 


DETERMINATION   OF   AN   ORBIT.  239 

In  the  case  of  the  earth, 


sm(Q"—  Q)' 
sm(O"— O') 


inCO"—  O)' 
and  the  complete  values  of  P  and  Q  become 

_    ^'sinCQ'-G) 
~f""         '' 


,3  /  ER'  sin(0'-  Q)  +  RR'  Bin(0"-  Q') 
~        '   \  '"  ; 


and  since  the  approximate  values 

P=C  Q  =  rr" 

differ  but  little  from  these,  as  will  appear  from  the  equations  (27)3, 
there  will  be  one  root  of  equation  (43)  which  gives  zr  nearly  equal 
to  180°  —  ij/.  This  root,  however,  cannot  satisfy  the  physical  con- 
ditions of  the  problem,  which  will  require  that  the  rays  of  light  in 
coming  from  the  planet  or  comet  to  the  earth  shall  proceed  from 
points  which  are  at  a  considerable  distance  from  the  eye  of  the 
observer.  Further,  the  negative  values  of  sin  z1  are  excluded  by  the 
nature  of  the  problem,  since  rf  must  be  positive,  or  z'  <  180°  ;  and 
of  the  three  positive  roots  which  may  result  from  equation  (43),  that 
being  excluded  which  gives  z'  very  nearly  equal  to  180°  —  ij/,  there 
will  remain  two,  of  which  one  will  be  excluded  if  it  gives  z'  greater 
than  180°  —  ^r,  and  the  remaining  one  will  be  that  which  belongs 
to  the  orbit  of  the  planet  or  comet.  It  may  happen,  however,  that 
neither  of  these  two  roots  is  greater  than  180°  —  ^'j  in  which  case 
both  will  satisfy  the  physical  conditions  of  the  problem,  and  hence 
the  observations  will  be  satisfied  by  two  wholly  different  systems  of 
Clements.  It  will  then  be  necessary  to  compare  the  elements  com- 
puted from  each  of  the  two  values  of  z'  with  other  observations  in 
order  to  decide  which  actually  belongs  to  the  body  observed. 

In  the  other  case,  in  which  cos  £  is  negative,  the  negative  roots 
being  excluded  by  the  condition  that  r'  is  positive,  the  positive  root 
must  in  most  cases  belong  to  the  orbit  of  the  earth,  and  the  three 
observations  do  not  then  belong  to  the  same  body.  However,  in  the 
case  of  the  orbit  of  a  comet,  when  the  eccentricity  is  large,  and  the 
intervals  between  the  observations  are  of  considerable  magnitude,  if 


240  THEORETICAL   ASTRONOMY. 

the  approximate  values  of  P  and  Q  are  computed  directly,  by  means 
of  approximate  elements  already  known,  from  the  equations 

p  _    r/  sin  (ur  —  u) 

-/^sirTK^')' 

^  _  <ws  /  ">  sin  (uf-  u)  +  *V'  sin  (u"-  u'} 
-2r  l 


it  may  occur  that  cos  £  is  negative,  and  the  positive  root  will  actually 
belong  to  the  orbit  of  the  comet.  The  condition  that  one  value  of 
zf  shall  be  very  nearly  equal  to  180°  —  $',  requires  that  the  adopted 
values  of  P  and  Q  shall  differ  but  little  from  those  derived  directly 
from  the  places  of  the  earth  ;  and  in  the  case  of  orbits  of  small 
eccentricity  this  condition  will  always  be  fulfilled,  unless  the  intervals 
between  the  observations  and  the  distance  of  the  planet  from  the  sun 
are  both  very  great.  But  if  the  eccentricity  is  large,  the  difference 
may  be  such  that  no  root  will  correspond  to  the  orbit  of  the  earth. 

84.  We  may  find  an  expression  for  the  limiting  values  of  mc  and 
£,  within  which  equation  (43)  has  four  real  roots,  and  beyond  which 
there  are  only  two,  one  positive  and  one  negative.  This  change  in 
the  number  of  real  roots  will  take  place  when  there  are  two  equal 
roots,  and,  consequently,  if  we  proceed  under  the  supposition  that 
equation  (43)  has  two  equal  roots,  and  find  the  values  of  m0  and  £ 
which  will  accord  with  this  supposition,  we  may  determine  the  limits 
required. 

"Differentiating  equation  (43)  with  respect  to  zf,  we  get 

cos  (d  —  C)  =  4m0  sin  V  cos  z'  ; 

and,  in  the  case  of  equal  roots,  the  value  of  zf  as  derived  from  this 
must  also  satisfy  the  original  equation 

sin  (z'  —  C)  =  m0  sin  V. 

To  find  the  values  of  m0  and  £  which  will  fulfil  this  condition,  if  we 
eliminate  m0  between  these  equations,  we  have 

sin  z'  cos  (z*  —  C)  =  4  cos  z'  sin  (z'  —  C), 
from  which  we  easily  find 

sin  (2z'  —  0  =  1  sin  C.  (45) 

This  gives  the  value  of  £  in  terms  of  z'  for  which  equation  (43)  haa 


DETERMINATION   OF   AN   ORBIT.  241 

equal  roots,  and  at  which  it  ceases  to  have  four  real  roots.  To  find 
the  corresponding  expression  for  m0,  we  have 

_  sin  (Y  —  C) cos  (Y  —  C) 

0  ;        sin  4z'        ~  4  sin  V  cos  / 

in  which  we  must  use  the  value  of  £  given  by  the  preceding  equation. 
Now,  since  sin  (2zf  —  £)  must  be  within  the  limits  —  1  and  -f-  1,  the 
limiting  values  of  sin  £  will  be  +  f  and  — |,  or  £  must  be  within  the 
limits  +  36°  52'.2  and  —  36°  52'.2,  or  143°  7'.8  and  216°  52'.2.  If 
£  is  not  contained  within  these  limits,  the  equation  cannot  have  equal 
roots,  whatever  may  be  the  value  of  ra0,  and  hence  there  can  only  be 
two  real  roots,  of  which  one  will  be  positive  and  one  negative.  If 
for  a  given  value  of  £  we  compute  z'  from  equation  (45),  and  call 
this  ZQJ  or 

sin  O0'  —  C)  =  |  sin  C, 

we  may  find  the  limits  of  the  values  of  m0,  within  which  equation 
(43)  has  four  real  roots.  The  equation  for  ZQ'  will  be  satisfied  by 
the  values 

2*0'-C,  180°-(2z0'-C); 

and  hence  there  will  be  two  values  of  m0,  which  we  will  denote  by 
ml  and  m2,  for  which,  with  a  given  value  of  £,  equation  (43)  will 
have  equal  roots.  Thus  we  shall  have 

m=sin(y-C) 
sin  \f      ' 

and,  putting  in  this  equation  180°  —  (2z/  —  £)  instead  of  2z0'  —  £,  or 
90°  —  (V  —  C)  in  place  of  z0', 

cos< 


It  follows,  therefore,  that  for  any  given  value  of  f,  if  m0  is  not 
within  the  limits  assigned  by  the  values  of  mx  and  m2,  equation  (43) 
will  only  have  two  real  roots,  one  positive  and  one  negative,  of 
which  the  latter  is  excluded  by  the  nature  of  the  problem,  and  the 
former  may  belong  to  the  orbit  of  the  earth.  But  if  P  and  Q  differ 
so  much  from  their  values  in  the  case  of  the  orbit  of  the  earth  that 
z1  is  not  very  nearly  equal  to  180°  — tj/,  the  positive  root,  when  £ 
exceeds  the  limits  -f  36°  52'.2  and  —36°  52'.2,  may  actually  satisfy 
the  conditions  of  the  problem,  and  belong  to  the  orbit  of  the  body 
observed. 

16 


242  THEORETICAL   ASTRONOMY. 

When  C  is  within  the  limits  143°  7'.8  and  216°  52'.2,  there  will 
be  four  real  roots,  one  positive  and  three  negative,  if  m0  is  within  the 
limits  ml  and  m2  ;  but,  if  m0  surpasses  these  limits,  there  will  be  only 
two  real  roots. 

Table  XII.  contains  for  values  of  £  from  —  36°  52'.2  to  +  36°  52'.2 
the  values  of  ml  and  m2,  and  also  the  values  of  the  four  real  roots 
corresponding  respectively  to  m1  and  m2. 

In  every  case  in  which  equation  (43)  has  three  positive  roots  and 
one  negative  root,  the  value  of  m0  must  be  within  the  limits  indicated 
by  m1  and  m2,  and  the  values  of  zf  will  be  within  the  limits  indicated 
by  the  quantities  corresponding  to  ml  and  m2  for  each  root,  which 
we  designate  respectively  by  z/,  z/,  z3',  and  z/.  The  table  will  show, 
from  the  given  values  of  m0  and  180°  —  ij/,  whether  the  problem 
admits  of  two  distinct  solutions,  since,  excluding  the  value  of  z', 
which  is  nearly  equal  to  180°  —  ij/,  and  corresponds  to  the  orbit  of 
the  earth,  and  also  that  which  exceeds  180°,  it  will  appear  at  once 
whether  one  or  both  of  the  remaining  two  values  of  z1  will  satisfy 
the  condition  that  z'  shall  be  less  than  180°  —  <$,'.  The  table  will 
also  indicate  an  approximate  value  of  z',  by  means  of  which  the 
equation  (43)  may  be  solved  by  a  few  trials. 

For  the  root  of  the  equation  (43)  which  corresponds  to  the  orbit 
of  the  earth,  we  have  pf  =  0,  and  hence  from  (36)  we  derive 


Substituting  this  value  for  k0  in  the  general  equation  (32),  we  have 

—  —  - 

and,  since  pr  must  be  positive,  the  algebraic  sign  of  the  numerical 
value  of  1Q  will  indicate  whether  rf  is  greater  or  less  than  Rf.  It  is 
easily  seen,  from  the  formulae  for  lw  b,  d,  &c.,  that  in  the  actual 
application  of  these  formulae,  the  intervals  between  the  observations 
not  being  very  large,  10  will  be  positive  when  /9'  —  /?0  and  sin  (O'  —  K) 
have  contrary  signs,  and  negative  when  /9r  —  /90  has  the  same  sign  as 
sin  (O'  —  jfif).  Hence,  when  O'  —  K  is  less  than  180°,  rr  must  be 
less  than  Rr  if  ft'  —  /90  is  positive,  but  greater  than  Rf  if  /?'  —  /90  is 
negative.  When  O'  —  K  exceeds  180°,  rr  will  be  greater  than  R' 
if  /9'  —  /?0  is  positive,  and  less  than  &'  if  /?'  —  /90  is  negative.  We 
may,  therefore,  by  means  of  a  celestial  globe,  determine  by  inspection 
whether  the  distance  of  a  comet  from  the  sun  is  greater  or  less  than 


DETERMINATION   OF   AN   ORBIT.  243 

that  of  the  earth  from  the  sun.  Thus,  if  we  pass  a  great  circ'e 
through  the  two  extreme  observed  places  of  the  comet,  rf  must  be 
greater  than  R'  when  the  place  of  the  comet  for  the  middle  observa- 
tion is  on  the  same  side  of  this  great  circle  as  the  point  of  the 
ecliptic  which  corresponds  to  the  place  of  the  sun.  But  when  the 
middle  place  and  the  point  of  the  ecliptic  corresponding  to  the  place 
of  the  sun  are  on  opposite  sides  of  the  great  circle  passing  through 
the  first  and  third  places  of  the  comet,  r'  must  be  less  than  R'. 

85.  From  the  values  of  //  and  r'  derived  from  the  assumed  values 

T" 
P  =  —  and   Q  =  rr",  we  may  evidently  derive  more  approximate 

values  of  these  quantities,  and  thus,  by  a  repetition  of  the  calcula- 
tion, make  a  still  closer  approximation  to  the  true  value  of  pr.  To 
derive  other  expressions  for  P  and  Q  which  are  exact,  provided  that 
rf  and  pr  are  accurately  known,  let  us  denote  by  sff  the  ratio  of  the 
sector  of  the  orbit  included  by  r  and  r'  to  the  triangle  included  by 
the  same  radii-vectores  and  the  chord  joining  the  first  and  second 
places  ;  by  s'  the  same  ratio  with  respect  to  r  and  r",  and  by  s  this 
ratio  with  respect  to  r'  and  r".  These  ratios  s,  sf,  s"  must  neces- 
sarily be  greater  than  1,  since  every  part  of  the  orbit  is  concave 
toward  the  sun.  According  to  the  equation  (30)^  we  have  for  the 
areas  of  the  sectors,  neglecting  the  mass  of  the  body, 


and  therefore  we  obtain 

s"[r/]=:TV£>  /[W'J^rV^  s[tV']=Ty$.     (46) 

Then,  since 


we  shall  have 
and,  consequently, 


[r/] 

- 


..          T        8  f  ,  _^ 

==-  (47) 


T-T"  /  oV  <?</  QQ"  \  ^       ' 

—        I  £l_  4.  2  __  !*    I  9r'8 
+  "' 


Substituting  for  s,  s',  and  s"  their  values  from  (46),  we  have 

["-']-C^"]  «« 

.  [r/']  .  [r'r"]     '  «r"  ' 


244  THEORETICAL   ASTRONOMY. 

The  angular  distance  between  the  perihelion  and  node  being  denoted 
by  to,  the  polar  equation  of  the  conic  section  gives 


7) 

—  =  1  -f  e  cos  (u  —  w), 


-  =  1  -f  e  cos  (u"  —  01). 


(50) 


If  we  multiply  the  first  of  these  equations  by  sin  (u"—  u'\  the  second 
by  —  sin  (uft  —  u\  and  the  third  by  sin  (uf  —  u),  add  the  products 
and  reduce,  we  get 

P-  sin  (u"  —  O  —  £  sin  (u"  —  u)+^,  sin  (u'  —  u)  =  sin  (u"  —  u') 

—  sin  (u"  —  u)  -|-  sin  (ur  —  u)  ; 
and,  since 

sin  (u"  —  u')  =  2  sin  J  (w"  —  1/)  cos  J  (w"  —  w'), 

sin  (u"—  u)  —  sin  (u'—  u)  =  2  sin  |  (u"—  u')  cos  |  (w"  -f  u'—  2w), 

the  second  member  reduces  to 

4  sin  i  (u"  —  «')  sin  J  (w"  —  w)  sin  J  (wf  —  w\ 
Therefore,  we  shall  have 

4rrV"  sin  \  (u"  —  u'}  sin  j  (uy/  —  u)  sin  j  (uf  —  u) 
^  r-  //'  sin  (w"_  w')  _  rr»  sin  (w//_  w)  ^_  r/  sin  ^/_  tty 

If  we  multiply  both  numerator  and  denominator  of  this  expression 
by 

2rr'r"  cos  i  (u"  —  it')  cos  J  («"  —  u)  cos  J  (w;  —  u), 

it  becomes,  introducing  [rr'],  [rr/r],  and  [r'r"], 

1 


__  __ 
'  "     [rV'j  +  [r/]  —  [rr"]  '  2nV'  cos  1  (M"—  M')  cos  J  (it"—  w)  cos  J  (*'—  w)" 

Substituting  this  value  of  p  in  equation  (49),  it  reduces  to 

rr"  r'2 


'  rr"  cos  J  (M"  —  t*')  cos  I  («"  —  w)  cos  j  (wr  —  w)' 


(51) 

86.  If  we  compare  the  equations  (47)  with  the  formula  (28)3,  we 
derive 


DETERMINATION   OF   AN   ORBIT.  245 

Consequently,  in  the  first  approximation,  we  may  take 


If  the  intervals  of  the  times  are  not  very  unequal,  this  assumption 
will  differ  from  the  truth  only  in  terms  of  the  third  order  with  respect 
to  the  time,  and  in  terms  of  the  fourth  order  if  the  intervals  are 
equal,  as  has  already  been  shown.  Hence,  we  adopt  for  the  first 
approximation, 


the  values  of  r  and  t"  being  computed  from  the  uncorrected  times 
of  observation,  which  may  be  denoted  by  t0,  t0f,  and  tQ".  With  the 
values  of  P  and  Q  thus  found,  we  compute  rr,  and  from  this  pf,  p, 
and  p",  by  means  of  the  formulae  already  derived. 

The  heliocentric  places  for  the  first  and  third  observations  may 
now  be  found  from  the  formulae  (71)3  and  (72)3,  and  then  the  angle 
u"  —  u  between  the  radii-vectores  r  and  r"  may  be  obtained  in 
various  ways,  precisely  as  the  distance  between  two  points  on  the 
celestial  sphere  is  obtained  from  the  spherical  co-ordinates  of  these 
points.  When  u"  —  u  has  been  found,  we  have 

MA* 

sin  (u"  —  u'}  =  -—  sin  (u"  —  u), 

n"r"  (53) 

•  Xf  -v  IV       I  •///  \ 

sin  (u  —  u)  =  —  —  sin  (u  —  u), 

from  which  u"  —  uf  and  uf  —  u  may  be  computed.  From  these 
results  the  ratios  s  and  s"  may  be  computed,  and  then  new  and  more 
approximate  values  of  P  and  Q.  The  value  of  u"  —  u,  found  by 
taking  the  sum  of  u"  —  ur  and  uf  —  it  as  derived  from  (53),  should 
agree  with  that  used  in  the  second  members  of  these  equations, 
within  the  limits  of  the  errors  which  may  be  attributed  to  the 
logarithmic  tables. 

The  most  advantageous  method  of  obtaining  the  angles  between 
the  radii-vectores  is  to  find  the  position  of  the  plane  of  the  orbit 
directly  from  I,  lfr,  6,  and  6",  and  then  compute  it,  u',  and  u"  directly 
from  Q,  and  i,  according  to  the  first  of  equations  (82)!.  It  will  be 
expedient  also  to  compute  rf,  I'  and  br  from  p',  ^',  and  /?',  and  the 
agreement  of  the  value  of  r',  thus  found,  with  that  already  obtained 
from  equation  (37),  will  check  the  accuracy  of  part  of  the  numerical 


THEORETICAL   ASTRONOMY. 

calculation.  Further,  since  the  three  places  of  the  body  must  be  in 
a  plane  passing  through  the  centre  of  the  sun,  whether  P  and  §  are 
exact  or  only  approximate,  we  must  also  have 

tan  bf  =  tan  i  sin  (f  —  &  ), 

and  the  value  of  b'  derived  from  this  equation  must  agree  with  that 
computed  directly  from  /?',  or  at  least  the  difference  should  not  exceed 
what  may  be  due  to  the  unavoidable  errors  of  logarithmic  calcula- 
tion. 

We  may  now  compute  n  and  n"  directly  from  the  equations 

_r'r"  sin  (?/'—<)  „_  rr'  sm(u'—u) 

~  rr"  sin  (u"—  u)  '  ~  rr"  sin  (u"—  u)  ' 

but  when  the  values  of  u,  uf,  and  u"  are  those  which  result  from  the 
assumed  values  of  P  and  §,  the  resulting  values  of  n  and  n"  will 
only  satisfy  the  condition  that  the  plane  of  the  orbit  passes  through 
the  centre  of  the  sun.  If  substituted  in  the  equations  (29),  they  will 
only  reproduce  the  assumed  values  of  P  and  §,  from  which  they 
have  been  derived,  and  hence  they  cannot  be  used  to  correct  them. 
If,  therefore,  the  numerical  calculation  be  correct,  the  values  of  n 
and  n"  obtained  from  (54)  must  agree  with  those  derived  from  equa- 
tions (31),  within  the  limits  of  accuracy  admitted  by  the  logarithmic 
tables. 

The  differences  u"  —  u'  and  ur  —  u  will  usually  be  small,  and 
hence  a  small  error  in  either  of  these  quantities  may  considerably 
affect  the  resulting  values  of  n  and  n".  In  order  to  determine 
whether  the  error  of  calculation  is  within  the  limits  to  be  expected 
from  the  logarithmic  tables  used,  if  we  take  the  logarithms  of  both 
members  of  the  equations  (54)  and  differentiate,  supposing  only  n, 
n",  and  uf  to  vary,  we  get 


=  —  cot  (it"  —  u')du', 
d  logen"  =  -j-  cot  (V  —  u)  du'. 

Multiplying  these  by  0.434294,  the  modulus  of  the  common  system 
of  logarithms,  and  expressing  duf  in  seconds  of  arc,  we  find,  in  units 
of  the  seventh  decimal  place  of  common  logarithms, 


d  log  n  =  —  21.055  cot  (u"  —  u')  du', 
d  log  n"  =  +  21.055  cot  (u'  —  u)  du'. 

If  we  substitute  in  these  the  differences  between  log  n  and  log  n"  as 
found  from  the  equations  (54),  and  the  values  already  obtained  by 


DETERMINATION   OF   AN   ORBIT.  247 

means  of  (31),  the  two  resulting  values  of  du'  should  agree,  and  the 
magnitude  of  du'  itself  will  show  whether  the  error  of  calculation 
exceeds  the  unavoidable  errors  due  to  the  limited  extent  of  the 
logarithmic  tables.  When  the  agreement  of  the  two  results  for  n 
and  n"  is  in  accordance  with  these  conditions,  and  no  error  has  been 
made  in  computing  n  and  n"  from  P  and  Q  by  means  of  the  equa- 
tions (31),  the  accuracy  of  the  entire  calculation,  both  of  the  quan- 
tities which  depend  on  the  assumed  values  of  P  and  Q,  and  of  those 
which  are  obtained  independently  from  the  data  furnished  by  observa- 
tion, is  completely  proved. 

87.  Since  the  values  of  n  and  n"  derived  from  equations  (54) 
cannot  be  used  to  correct  the  assumed  values  of  P  and  §,  from 
which  r,  rf,  u,  uf,  &c.  have  been  computed,  it  is  evidently  necessary 
to  compute  the  values  for  a  second  approximation  by  means  of  the 
series  given  by  the  equations  (26)3,  or  by  means  of  the  ratios  s  and 
s".  The  expressions  for  n  and  n"  arranged  in  a  series  with  respect 
to  the  time  involve  the  differential  coefficients  of  rf  with  respect  to  t} 
and,  since  these  are  necessarily  unknown,  and  cannot  be  conveniently 
determined,  it  is  plain  that  if  the  ratios  s  and  s"  can  be  readily  found 
from  r,  r',  r/f,  u,  uf,  uff,  and  r,  r',  rr/,  so  as  to  involve  the  relation 
between  the  times  of  observation  and  the  places  in  the  orbit,  they 
may  be  used  to  obtain  new  values  of  P  and  Q  by  means  of  equations 
(48)  and  (51),  to  be  used  in  a  second  approximation. 

Let  us  now  resume  the  equation 

M=E  —  esinE, 


a 
and  also  for  the  third  place 


Subtracting,  we  get 

^  =  E"  —  E  —  2e  sin  J  (E"  —  E}  cos  J  (E"  +  JE).  (55; 

«2 

This  equation  contains  three  unknown  quantities,  a,  e,  and  the  dif- 
ference E"  —  E.  We  can,  however,  by  means  of  expressions  in- 
volving r,  rffj  Uj  and  it",  eliminate  a  and  e.  Thus,  since  p  =  a(l  —  e2), 
we  have 


=  oVl^T1  (E"  -  E  -  2e  sin  £  (E"  -  E)  cos  J  (E"  +  E}).  (56) 


248  THEORETICAL   ASTRONOMY. 

From  the  equations 

V/r  sin  &  =  T/g(l-fe)  sin  %E,  V^_  sin  &'  =  l/a(l  +  e)  sin  £ £", 

Vr  cos Av  =  1/0(1—  e)  cos  J^,  T/r"  cos  jw"  =  l/o(l— e)  cos £.£", 

since  t?"  —  t?  =  it"  —  it,  we  easily  derive 

Vr?  sin  i  (w"  —  u)  =  aVl—e*  sin  J  (£"  —  E),  (57) 

and  also 

a  cos  J  (.£"  —  E}  —  ae  cos  £  (En  +  E}  =  VW'  cos  \  (u"  —  u), 


or 

e  cos  |  (E"  +  JE)  =  cos  £  QE"  -  £)  -  y  "    UUB^ v"  ~  ";.     (58) 

Substituting  this  value  of  ecos|(^;/+  E)  in  equation  (56),  we  get 


r'p  =  a2lr=72  (£"  —  E—sm  (E"  —  E)) 

+  2al/1^72  sin  £  (£"  —  E)  cos  J  (M"  —  u)  VW', 

and  substituting,  in  the  last  term  of  this,  for  al/1  —  e2,  its  value  from 
(57),  the  result  is 


=  a*iT=#  (E"  —  E  —  sin  (E''  —  E))  +  rr"  sin  (u"  —  u).   (59) 
From  (57)  we  obtain 

*       "--> 


p         sn 
or 


_  .      /      rr"sin(tt"—  M)      \»  _  1 

=  \2v/r/7cosJ(w"—  u)l  p  sin*  %(£"— 

Therefore,  the  equation  (59)  becomes 
1    £»  _  E-  sin  £"- 


C      ]'(     J 

Let  Jtr  be  the  chord  of  the  orbit  between  the  first  and  third  places, 
and  we  shall  have 


x'2  =  (r  +  r")2  —  4rr"  cos2  J  (n"  —  w). 
Now,  since  the  chord  #'  can  never  exceed  r  +  r/r,  we  may  put 

x'=(r-fr")sinr',  (61) 

and  from  this,  in  combination  with  the  preceding  equation,  we  derive 
7  cos  J  (M"  —  M)  =  (r  +  /')  cos  /.  (62) 


DETERMINATION   OF   AN   ORBIT.  249 

Substituting   this   value,   and   \rr"~\  =  -,  Vp,   in   equation    (60),   it 
reduces  to 

E"-E-Sm(E"-E)  r"  1        1  _ 

sin3  1  (E"  —  E)        "(r+  r")8  cos3  p  *  7*  "*"  7 

The  elements  a  and  e  are  thus  eliminated,  but  the  resulting  equation 
involves  still  the  unknown  quantities  E"  —  E  and  sf.  It  is  neces- 
sary, therefore,  to  derive  an  additional  equation  involving  the  same 
unknown  quantities  in  order  that  E"  —  E  maybe  eliminated,  and 
that  thus  the  ratio  sr,  which  is  the  quantity  sought,  may  be  found. 
From  the  equations 

r  —  a  —  ae  cos  E,  r"  =  a  —  ae  cos  E", 

we  get 

r"  +  r  =  2a  —  2ae  cos  J  (E"  -f  E)  cos  |  (E"  —  E). 

Substituting  in  this  the  value  of  ecos%(E"-}-  E)  from  (58),  we  have 

r"  4.  r  =  2a  sin*  j  (E"  —  E)  +  21/77'  cos  j  (w';  —  t*)  cos  J  (^"  —  ^), 
and  substituting  for  sin  l(Etr—  E)  its  value  from  (57),  there  results 


But,  since 


t*"~  u)  (l-2sin 


r  sin'  j  (u;/  —  M)  _  (Kr])2  =  2ry2  /  _  1 

^>  ~  2prrff  cos2  ^  (t*"  —  u)~  s'2  \  21/^y7  cos 

we  have 


from  which  we  derive 


which  is  the  additional  equation  required,  involving  E"  —  j&and  «' 
as  unknown  quantities. 
Let  us  now  put 


(65) 


,       sin'V 
J="WF" 


250  THEORETICAL   ASTRONOMY. 

and  the  equations  (63)  and  (64)  become 


.- 
yf     s'3'"' 

(66) 


When  the  value  of  yf  is  known,  the  first  of  these  equations  will 
enable  us  to  determine  s',  and  hence  the  value  of  #',  or  sin2£(i£"  —  E}t 
from  the  last  equation. 

The  calculation  of  f  ma7  be  facilitated  by  the  introduction  of  an 
additional  auxiliary  quantity.  Thus,  let 


(67) 
and  from  (62)  we  find 

cos  /  =  cos  £  (u"  —  u)  .,  =  2  cos  |  (u"  —  u}  cos'/  tan  /, 

or 

cos  /  =  sin  2/  cos  £  (u"  —  u).  (68) 

We  have,  also, 

*'2  =  (r  +  r'J  —  4rr"  cos2  J  (t*"  —  «), 
which  gives 

x'2  =  (r  —  r")2  +  4rr"  sina  J  («*"  —  u}. 

Multiplying  this  equation  by  cos2J(it/r  —  u)  and  the  preceding  one 
by  sin2  \  (u"  —  u),  and  adding,  we  get 

x'2  =  (r  +  r")2  sin2  £  (w"  —  u)  +  (r  —  /r)a  cos2  ^  (wv  —  it).       (69) 
From  (67)  we  get 


and,  therefore, 

cos2/=L-=i 

r  -f-  r" 

so  that  equation  (69)  may  be  written 

_f2       =:  sin2  •/  =  sin2  £  (w/;  —  w)  +  cos2  2/  cos2  j  (u"  —  u). 

We  may,  therefore,  put 

sin  /  cos  G1  =  sin  ^  (w"  —  w), 

sin  /  sin  G'  =  cos  £  (w"  —  u)  cos  2/,  (70) 

cos  /  =  cos  I  («"  —  w)  sin  2/, 


DETERMINATION   OF   AN   ORBIT.  251 

from  which  p'  may  be  derived  by  means  of  its  tangent,  so  that  sin  ff 
shall  be  positive.  The  auxiliary  angle  G'  will  be  of  subsequent  use 
in  determining  the  elements  of  the  orbit  from  the  final  hypothesis  for 
P and  Q. 

88.  We  shall  now  consider  the  auxiliary  quantity  y'  introduced 
into  the  first  of  equations  (66).     For  brevity,  let  us  put 

and  we  shall  have 

,_  — sm  9 m 

2g  —  sin  2g 
This  gives,  by  differentiation, 

dy'       n  4  sin2  q  dq 

—4-  =  o  cot  q  do  —  » -. — g-i 

y  ^9  —  sm  *>9 

or 

di/ 

-f-  =  2>y  cot  g  —  4y 2  cosec  g. 

dg 

The  last  of  equations  (65)  gives  a/— sin2£#,  and  hence 

dg 

-—  =  2  cosec  g. 

Therefore  we  have 

dy'  _  Qyf  cos  g  —  8yfa  =  3  (1  -  2aQ  y'  —  4y" 
dx'  ~  sin2  g  2x'  (1  —  #') 

It  is  evident  that  we  may  expand  yf  into  a  series  arranged  in  refer- 
ence to  the  ascending  powers  of  x1 ',  so  that  we  shall  have 

y'  =.  a,  -f-  fix'  -f-  ?%'*  ~\-  ^'3  +  £^'*  -f-  C«'5  -f~  &C. 

Differentiating,  we  get 

and  substituting  for  -g*  the  value  already  obtained,  there  results 


afc-r  +  (4r  —  2/9)  ^'2  +  (6*  —  4r)  ^'3  +  (8e  —  65)  ^  +  (IOC  — 

=  (3a  —  4a2)  +  (3^  —  6a  —  8a/9)  a;'  +  (3r  —  6^  —  4/52  — 

+  (35  —  6r  —  8/?r  —  8a^)  x'3  -f  (3e  —  65  —  4f  —  8/?<5  —  8ae)  ^ 

4.  (3C  —  6--  —  8r<5  —  8/?e  —  8aC)  xf&  +  &c. 

Since  the  coefficients  of  like  powers  of  x'  must  be  equal,  we  nave 
3tt  _  4a2  =  o,  3{3  —  6a  —  8a£  =  2A 

3r  —  6,5  —  4;52  —  8<*r  =  2  (2r  —  £),  &c. ; 


252  THEORETICAL   ASTRONOMY. 

and  hence  we  derive 

«  =  i,  P  =  —  T9o,  r=T§-5,  d  =  T 

cr  _      6228  f  _      265896  «,  _         191390 

e  —  —  — 


336875* 


_      265896 
—  2TS3e573» 


Therefore  we  have 


If  we  multiply  through  by  V,  and  put 

*"  +  dii-fts*'5 

<=.,  (72; 


we  obtain 

Combining  this  with  the  second  of  equations  (66),  the  result  is 

VY+£= §+/+?'. 

If  we  put 

V  =  __£__,  (74) 

we  shall  have 


But  from  the  first  of  equations  (66)  we  get 

7  -*(/-i;; 

and  therefore  we  have 

^^-  (75) 

•  H-  ^ 

As  soon  as  3/  is  known,  this  equation  will  give  the  corresponding 
value  of  s'. 

Since  £'  is  a  quantity  of  the  fourth  order  in  reference  to  the  differ- 
ence |  (E"  —  E\  we  may  evidently,  for  a  first  approximation  to  the 
value  of  yf,  take 


and  with  this  find  sf  from  (75),  and  the  corresponding  value  of  x* 
from  the  last  of  equations  (66).  With  this  value  of  xf  we  find  the 
corresponding  value  of  £r,  and  recompute  ?/,  s',  and  xf  ;  and,  if  tL« 


DETERMINATION   OF   AN   ORBIT.  253 

value  of  £'  derived  from  the  last  value  of  xf  differs  from  that  already 
used,  the  operation  must  be  repeated. 

It  will  be  observed  that  the  series  (72)  for  £'  converges  with  great 
rapidity,  and  that  for  E"  —  j£=94°  the  term  containing  #/6  amounts 
to  only  one  unit  of  the  seventh  decimal  place  in  the  value  of  c'.  Table 
XIV.  gives  the  values  of  £'  corresponding  to  values  of  xf  from  0.0 
to  0.3,  or  from  E"  —  E=0  to  E"  —  E  =  132°  50'.6.  Should  a 
case  occur  in  which  E"  —  E  exceeds  this  limit,  the  expression 


*  ~  E"—E  —  sin  (E"—  E) 

may  then  be  computed  accurately  by  means  of  the  logarithmic  tables 
ordinarily  in  use.  An  approximate  value  of  x'  may  be  easily  found 
with  which  y'  may  be  computed  from  this  equation,  and  then  £'  from 
(73).  With  the  value  of  £'  thus  found,  if  may  be  computed  from 
(74),  and  thus  a  more  approximate  value  of  x'  is  immediately 
obtained. 

The  equation  (75)  is  of  the  third  degree,  and  has,  therefore,  three 
roots.  Since  s'  is  always  positive,  and  cannot  be  less  than  1,  it 
follows  from  this  equation  that  rf  is  always  a  positive  quantity.  The 
equation  may  be  written  thus  : 

8*—  a"  —  ,Y—   1,/=0, 

and  there  being  only  one  variation  of  sign,  there  can  be  only  one 
positive  root,  which  is  the  one  to  be  adopted,  the  negative  roots  being 
excluded  by  the  nature  of  the  problem.  Table  XIII.  gives  the 
values  of  logs'2  corresponding  to  values  of  if  from  j/=0  to  J/=0.6. 
When  r/  exceeds  the  value  0.6,  the  value  of  sf  must  be  found  directly 
from  the  equation  (75). 

89.  We  are  now  enabled  to  determine  whether  the  orbit  is  an 
ellipse,  parabola,  or  hyperbola.  In  the  ellipse  x  =  sin*  \(E"  —  E) 
is  positive.  In  the  parabola  the  eccentric  anomaly  is  zero,  and  hence 
x  =  0.  In  the  hyperbola  the  angle  which  we  call  the  eccentric 
anomaly,  in  the  case  of  elliptic  motion,  becomes  imaginary,  and 
hence,  since  sin  \  (E"  —  E)  will  be  imaginary,  x'  must  be  negative. 
It  follows,  therefore,  that  if  the  value  of  x'  derived  from  the  equa- 
tion 


is  positive,  the  orbit  is  an  ellipse  ;  if  equal  to  zero,  the  orbit  is  a 
parabola  ;  and  if  negative,  it  is  a  hyperbola. 


254  THEORETICAL,   ASTRONOMY. 

For  the  case  of  parabolic  motion  we  have  xr  —  0,  and  the  second 
of  equations  (66)  gives 

«"  =  j'  (76) 

If  we  eliminate  sr  by  means  of  both  equations,  since,  in  this  case, 
y'  =  f,  we  get 

m'l  =/*  -f-  |/l 

Substituting  in  this  the  values  of  m  and  I  given  by  (65),  we  obtain 

q    f 

—  —  —  =  3  sin  y  cos  rr  +  4  sin3  1/, 

which  gives 

6  ' 
—  =  6  sin  i/  cos2  tf  +  2  sin"  i/, 


or 

_f  T     ,  =  (sin  J/  +  cos  i/)8  +  (sin  i/  -  cos  tf?. 

This  may  evidently  be  written 


=         sn       +     -  sn 


the  upper  sign  being  used  when  f  is  less  than  90°,  and  the  lower 
sign  when  it  exceeds  90°.  Multiplying  through  by  (r  -j-  r/;)^,  and 
replacing  (r  -f-  r")  sin  ^  by  X,  we  obtain 

6r'  =  (r  +  r"  +  x)f  qz  (r  +  r"  -  x)f, 

which  is  identical  with  the  equation  (5G)3  for  the  special  case  of 
parabolic  motion. 

Since  x'  is  negative  in  the  case  of  hyperbolic  motion,  the  value  of 
£'  determined  by  the  series  (72)  will  be  different  from  that  in  the 
case  of  elliptic  motion.  Table  XIV.  gives  the  value  of  £'  corre- 
sponding to  both  forms;  but  when  x'  exceeds  the  limits  of  this  table, 
it  will  be  necessary,  in  the  case  of  the  hyperbola  also,  to  compute  the 
value  of  £'  directly,  using  additional  terms  of  the  series,  or  we  may 
modify  the  expression  for  y'  in  terms  of  Eff  and  E  so  as  to  be 
applicable. 

If  we  compare  equations  (44)!  and  (56)1?  we  get 

tan   E=  l/^I  tan   F; 


DETERMINATION   OF   AN   ORBIT.  255 

and  hence,  from  (58)w 


We  have,  also,  by  comparing  (65)!  with  (41  )w  since  a  is  negative  m 
the  hyperbola, 


2<7      ' 

which  gives 


Now,  since 

c< 

in  which  e  is  the  base  of  Naperian  logarithms,  we  have 

E  1/^1  =  loge  (cos  E  -f  1/^T  sin  E\ 
which  reduces  to 


or 

E=  I/HI  log.  <r. 

By  means  of  these  relations  between  jEJ  and  <r,  the  expression  for  yx 
may  be  transformed  so  as  not  to  involve  imaginary  quantities.  Thus 
we  have 

E"-E=  (loge  *"  -  loge  *)  V^=l  =  l/11 

sin  (^"  —  E)  =  sin  jK"  cos  E  —  cos  J£"  sin  E  =  a 


Zffa 

From  the  value  of  cos  E  we  easily  derive 

sin  \E  =  ^^  l/^l,  cos       = 

and  hence 


Therefore  the  expression  for  i/r  becomes 


'--4 

a 


256  THEORETICAL   ASTRONOMY. 

Since  the  auxiliary  quantity  a  in  the  hyperbola  is  always  positive, 
let  us  now  put 


and  we  have 

ic-i-a* 

/  =  --  —  --  (77) 


from  which  y'  may  be  derived  when  A  is  known. 
We  have,  further, 


sin" 
and  therefore 

*=-  ":.;-^"y-  (78, 

or 


These  expressions  for  y'  and  #'  enable  us  to  find  £'  when  x'  exceeds 
the  limits  of  the  table.  Thus,  we  obtain  an  approximate  value  of  x1 
by  putting 


from  which  we  first  find  sf  and  then  x'  from  the  second  of  equations 
(66).     Then  we  compute  A  from  the  formula  (79),  which  gives 


A  =  l  —  2xf  +  2lz'2  —  of,  (80) 

yr  from  (77),  and  £'  from  (73).  A  repetition  of  the  calculation,  using 
the  value  of  £'  thus  found,  will  give  a  still  closer  approximation  to 
the  correct  values  of  x'  and  s'  •  and  this  process  should  be  continued 
until  £'  remains  unchanged. 

90.  The  formulae  for  the  calculation  of  sf  will  evidently  give  the 
value  of  s  if  we  use  r,  r',  r",  uf,  and  u",  the  necessary  changes  in  the 
notation  being  indicated  at  once;  and  in  the  same  manner  using  T"  ', 
r,  rf,  u,  and  uf,  we  obtain  s".  From  the  values  of  «  and  s"  thus 
found,  more  accurate  values  of  P  and  Q  may  be  computed  by  means 
of  the  equations  (48)  and  (51).  We  may  remark,  however,  that  if 
!:he  times  of  the  observations  have  not  been  already  corrected  for  the 


DETERMINATION   OF   AN   ORBIT.  257 

time  of  aberration,  as  in  the  case  of  the  determination  of  an  unknown 
orbit,  this  correction  may  now  be  applied  as  determined  by  means  of 
the  values  of  p,  p',  and  p"  already  obtained.  Thus,  if  t0,  t0',  and  t0" 
are  the  uncorrected  times  of  observation,  the  corrected  values  will  be 

t  =  t0  —  Cp  sec  /?, 

t'  =tQ'  —  Cp'Becp,  (81) 

r  =  V—  0»"sec/9", 

in  which  log  C=  7.760523,  expressed  in  parts  of  a  day;  and  from 
these  values  of  ty  tfj  t"  we  recompute  r,  rr,  and  r",  which  values  will 
require  no  further  correction,  since  p,  p',  and  p",  derived  from  the 
first  approximation,  are  sufficient  for  this  purpose.  With  the  new 
values  of  P  and  Q  we  recompute  r,  rr,  r",  and  u,  uf,  u"  as  before, 
and  thence  again  P  and  §,  and  if  the  last  values  differ  from  the  pre- 
ceding, we  proceed  in  the  same  manner  to  a  third  approximation, 
which  will  usually  be  sufficient  unless  the  interval  of  time  between 
the  extreme  observations  is  considerable.  If  it  be  found  necessary 
to  proceed  further  with  the  approximations  to  P  and  Q  after  the 
calculation  of  these  quantities  in  the  third  approximation  has  been 
effected,  instead  of  employing  these  directly  for  the  next  trial,  we 
may  derive  more  accurate  values  from  those  already  obtained.  Thus, 
let  x  and  y  be  the  true  values  of  P  and  §  respectively,  with  which, 
if  the  calculation  be  repeated,  we  should  derive  the  same  values  again. 
Let  A#  and  &y  be  the  differences  between  any  assumed  values  of  x 
and  y  and  the  true  values,  or 


Then,  if  we  denote  by  #/,  yQf  the  values  which  result  by  direct  cal- 
culation from  the  assumed  values  x0  and  yw  we  shall  have 

xo  —  xo  =/GP<»  2/o)  =/GB  -f  Aa?,  y  +  Ay). 
Expanding  this  function,  we  get 

Ay  +  Ely*  +...., 


and  if  AO;  and  Ay  are  very  small,  we  may  neglect  terms  of  the  second 
order.  Further,  since  the  employment  of  x  and  y  will  reproduce  the 
same  values,  we  have 

/(*,2/)  =  0, 

and  hence,  since  A#  =  x0  —  x  and  &y  =  yQ  —  y, 


17 


258  THEORETICAL  ASTRONOMY. 

In  a  similar  manner,  we  obtain 


Let  us  now  denote  the  values  resulting  from  the  first  assumption  for 
P  and  Q  by  P,  and  §t,  those  resulting  from  P19  Ql  by  P2,  Q2,  and 
from  P2,  §2  by  P3,  §3;  and,  further,  let 

P1-P  =  a,  Pf-P1=a',  P3_P2_a", 

fc-e=ft,       e,-ft=ft',       e.-c,=yf. 

Then,  by  means  of  the  equations  for  x0f  —  XQ  and  y0f  —  yQ)  we  shall 
have 

a  =  A(P-z        B-y),  b  =  A  (P  -x)  +  B'  (Q  -y\ 

y-),  V  =  A(P-x)  +  ff^-y), 

y),  V'  =  A'(P,-x)+B(Q,-y\ 

If  we  eliminate  J.,  B,  Af,  and  P'  from  these  equations,  the  results 
are 

^       P(o'6"  —  q"6')  -f  P,  (0"6  —  0^)  +  P2  (oft'  —  a'6) 
(a'  J"  _  a"6')  +  (a"&  —  ab")  +  (06'  —  o'6) 

-  e  (o'y7  —  a;yy  )  +  Q,  <x'&  —  ^^o  +  $2  (^  —  g^) 

y  -         (afb"  —  a"b'}  +  ~(a"b  —  ab")  +  (06'  —  a'6)       ' 
from  which  we  get 

(a"  4.  a')  (a'b"—a"bf}  +  g"(V'6  —  «5'^ 


'ft"_  0"y  )  4.  (a"b  —  aft") 

/  r/        y 


~       3      (a'6"—  a"6')  +  (a"b  —  ab"}  +  (a6r—  a'6)" 

In  the  numerical  application  of  these  formulae  it  will  be  more  con- 
venient to  use,  instead  of  the  numbers  P,  P1?  P2,  §,  §1?  &c.,  the  loga- 
rithms  of  these  quantities,  so  that  a  =  log  Pl  —  log  P,  b  =  log  §t  —  log  Q, 
and  similarly  for  a'  ',  6',  ar/,  6r/,  —  which  may  also  be  expressed  in 
units  of  the  last  decimal  place  of  the  logarithms  employed,  —  and  we 
shall  thus  obtain  the  values  of  log  x  and  log  y.  With  these  values 
uf  log  x  and  log  y  for  log  P  and  log  Q  respectively,  we  proceed  with 
the  final  calculation  of  r,  rf,  r",  and  u9  uf,  u"  . 

When  the  eccentricity  is  small  and  the  intervals  of  time  between 
the  observations  are  not  very  great,  it  will  not  be  necessary  to  employ 
the  equations  (82)  ;  but  if  the  eccentricity  is  considerable,  and  if,  in 
addition  to  this,  the  intervals  are  large,  they  will  be  required.  It 
may  also  occur  that  the  values  of  P  and  Q  derived  from  the  last 
hypothesis  as  corrected  by  means  of  these  formulae,  will  differ  so 


DETERMINATION   OF   AN   ORBIT.  259 

much  from  the  values  found  for  x  and  y,  on  account  of  the  neglected 
terms  of  the  second  order,  that  it  will  be  necessary  to  recompute  these 
quantities,  using  these  last  values  of  P  and  Q  in  connection  with  the 
three  preceding  ones  in  the  numerical  solution  of  the  equations  (82). 

91.  It  remains  now  to  complete  the  determination  of  the  elements 
of  the  orbit  from  these  final  values  of  P  and  Q.  As  soon  as  & ,  i, 
and  u,  u',  u"  have  been  found,  the  remaining  elements  may  be  de- 
rived by  means  of  r,  rf,  and  u' —  u,  and  also  from  rfy  r",  and  u" — u' \ 
or,  which  is  better,  we  will  obtain  them  from  the  extreme  places,  and, 
if  the  approximation  to  P  and  Q  is  complete,  the  results  thus  found 
will  agree  with  those  resulting  from  the  combination  of  the  middle 
place  with  either  extreme. 

We  must,  therefore,  determine  a'  and  xf  from  r,  r",  and  u" — u, 
by  means  of  the  formulae  already  derived,  and  then,  from  the  second 
of  equations  (46),  we  have 


y-       —j~       -•> 

from  which  to  obtain  p.    If  we  compute  s  and  s"  also,  we  shall  have 
/  sr'r"  sin  (u"  —  u')\2      I  s"rrf  sin  (ur  —  u)\* 


and  the  mean  of  the  two  values  of  p  obtained  from  this  expression 
should  agree  with  that  found  from  (83),  thus  checking  the  calcula- 
tion and  showing  the  degree  of  accuracy  to  which  the  approximation 
to  P  and  Q  has  been  carried. 
The  last  of  equations  (65)  gives 

sm|(£"—  Jg?)  =  T/^,  (84) 

from  which  E"  —  E  may  be  computed.     Then,  from  equation  (57), 
since  e  =  sin  <p,  we  have 

•"""'=  ^ 

for  the  calculation  of  a  cos  <p.     But  p  =  a  (1  —  e2)  =  a  cos2  (p}  whence 

cos  r  =  -2—,  (86) 


which  may  be  used  to  determine  <p  when  e  is  very  nearly  equal   to 
unity;  and  then  e  may  be  found  from 

e  =  1  —  2  sin2  (45°  —  J<P). 


260  THEOKETICAL   ASTRONOMY. 

The  equations  (50)  give 

n\ 

e  cos  (u  —  to)  =  ±L  —  1, 

D 
e  cos  (u"  —  w)  =  ^  —  1, 

and  from  these,  by  addition  and  subtraction,  we  derive 

ajcoBi(nr'-ii)ooBG(ti"+  «)-«)=?  +  £- 

2e  sin  '  (u"  —  u)  sin  Q  (u"  +  «)—«)=?—     , 


(87) 


oy  means  of  which  e  and  a>  may  be  found. 
Since 

o  ,      r  —  r"  -    o  f 

cos  2/  =  —  ;  —  n,  sm  2y  =  —  ;  —  j 

r  +  r  '  r  +  r" 

we  have 

g    ,    1>  o^       _2^  o 

rTF  V/r//sin2/ 

p      p  _  2p  cot  2/ 

"  ' 


and  from  equations  (70), 

.  0  ,      sin  i  (it"  —  w)  tan  6rf  -of             cos  / 

cot  2/  ==  -  —  -  /  -  ,  sm  2/  =  -  T7-77  -  r. 

cos  /  cos  ^  (u  —  u) 

Therefore  the  formulae  (87)  reduce  to 

*  sin  «,  -      «"       t*     =  -  ==  tan  G' 


(88) 

«  cos  (w  —  £  (?/'  +  w))  =  -    x  _  —  sec  J  (wv  —  w;, 
cos/F  rr" 

from  which  also  6  and  co  may  be  derived.     Then 

sin  <p  =  e, 

and  the  agreement  of  cos  <p  as  derived  from  this  value  of  <p  with  that 
given  by  (86)  will  serve  as  a  further  proof  of  the  calculation.  The 
longitude  of  the  perihelion  will  be  given  by 

*  =  «+  £, 

or,  when  i  exceeds  90°,  and  the  distinction  of  retrograde  motion  is 
adopted,  by  K  =  &  —  co. 


DETERMINATION   OF   AN   OKBIT.  261 

To  find  a,  we  have 

p      _(a  cos  ^>)8 

~~ 


COS2  <f>  ~  p 

or  it  may  be  computed  directly  from  the  equation 


4s'W  cos2^  (u"  —  u)  sm*$  (E"  —  JB 


which  results  from  the  substitution,  in  the  last  term  of  the  preceding 
equation,  of  the  expressions  for  a  cos  <p  and  p  given  by  (83)  and  (85). 
Then  for  the  mean  daily  motion  we  have 


We  have  now  only  to  find  the  mean  anomaly  corresponding  to  any 
epoch,  and  the  elements  are  completely  determined.  For  the  true 
anomalies  we  have 

V  =  U  —  <i>,  Vf  =  U  —  to,  v"  =  u"  —  ct>  ' 

and  if  we  compute  r,  r'9  r"  from  these  by  means  of  the  polar  equa- 
tion of  the  conic  section,  the  results  should  agree  with  the  values  of 
the  same  quantities  previously  obtained.  According  to  the  equation 

(45)j,  we  have 

tan  \E  =  tan  (45°  —  £?)  tan  Jv, 

tan  \E'  =  tan  (45°  —  J?)  tan  jv',  (90) 

tan  ±E"  ==  tan  (45°  —  j?)  tan  W', 

from  which  to  find  E,  E'y  and  E"  .  The  difference  E"  —  E  should 
agree  with  that  derived  from  equation  (84)  within  the  limits  of 
accuracy  afforded  by  the  logarithmic  tables.  Then,  to  find  the  mean 

anomalies,  we  have 

M  =E  —  eswE, 

M'=E'—esmE',  (91; 

M"  =  E"—esmE"; 

and,  if  MQ  denotes  the  mean  anomaly  corresponding  to  any  epoch  Tt 
we  have,  also, 


?n  the  application  of  which  the  values  of  t,  t>  ',  and  t"  must  be  those 
which  have  been  corrected  for  the  time  of  aberration.     The  agree- 


262  THEORETICAL   ASTRONOMY. 

ment  of  the  three  values  of  M0  will  be  a  final  test  of  the  accuracy  of 
the  entire  calculation.  If  the  final  values  of  P  and  Q  are  exact, 
this  proof  will  be  complete  within  the  limits  of  accuracy  admitted 
by  the  logarithmic  tables. 

When  the  eccentricity  is  such  that  the  equations  (91)  cannot  be 
solved  with  the  requisite  degree  of  accuracy,  we  must  proceed  accord- 
ing to  the  methods  already  given  for  finding  the  time  from  the  peri- 
helion in  the  case  of  orbits  differing  but  little  from  the  parabola. 
For  this  purpose,  Tables  IX.  and  X.  will  be  employed.  As  soon  as 
v,  vf,  and  v"  have  been  determined,  we  may  find  the  auxiliary  angle 
V  for  each  observation  by  means  of  Table  IX. ;  and,  with  V  as  the 
argument,  the  quantities  M9  Mf ,  M"  (which  are  not  the  mean  anoma- 
lies) must  be  obtained  from  Table  VI.  Then,  the  perihelion  distance 
having  been  computed  from 

P 

we  shall  have 


in  which  log  (70  =  9.96012771,  for  the  determination  of  the  time  of 
perihelion  passage.  The  times  t,  t',  tu  must  be  those  which  have 
been  corrected  for  the  time  of  aberration,  and  the  agreement  of  the 
three  values  of  T  is  a  final  proof  of  the  numerical  calculation. 

If  Table  X.  is  used,  as  soon  as  the  true  anomalies  have  been  found, 
the  corresponding  values  of  log  B  and  log  C  must  be  derived  from 
the  table.  Then  w  is  computed  from 


and  similarly  for  wf  and  w"  ;  and,  with  these  as  arguments,  we  derive 
M,  Mr,  M"  from  Table  VI.     Finally,  we  have 

MBq*  _ 


(93) 

for  the  time  of  perihelion  passage,  the  value  of  O0  being  the  same  as 
in  (92). 

When  the  orbit  is  a  parabola,  e  =  I  and  p  —  2g,  and  the  elements 
Q  and  <o  can  be  derived  from  r,  r"9  u,  and  u"  by  means  of  the  equa- 


DETERMINATION   OF   AN   OEBIT.  263 

tions  (76),  (83),  and  (88),  or  by  means  of  the  formulae  already  given 
for  the  special  case  of  parabolic  motion. 

92.  Since  certain  quantities  which  are  real  in  the  ellipse  and  para- 
bola become  imaginary  in  the  case  of  the  hyperbola,  the  formulae 
already  given  for  determining  the  elements  from  r,  rn ',  u,  and  u" 
require  some  modification  when  applied  to  a  hyperbolic  orbit. 

When  sf  and  xf  have  been  found,  p,  e}  and  w  may  be  derived  from 
equations  (83)  and  (87)  or  (88)  precisely  as  in  the  case  of  an  elliptic 
orbit.  Since  xf  =  sin2  \  (E"  —  E),  we  easily  find 


sin  i  (E"  —  J£)  =  2  l/V  — 

and  equation  (85)  becomes 

sini(t*"—  u 

~ 


But  in  the  hyperbola  xf  is  negative,  and  hence  Vxr  —  xn  will  be 
imaginary  ;  and,  further,  comparing  the  values  of  p  in  the  ellipse 
and  hyperbola,  we  have  cos2^  =  —  tan2^,  or 

cos  <p  =  V  —  1  tan  4*. 
Therefore  the  equation  for  a  cos  <p  becomes 


if  a  is  considered  as  being   positive,  from  which  a  tan  if/  may  be 
obtained.     Then,  since  p  =  a  tan2  if/,  we  have 

tan  $  =  —f-  —  ,  (96) 


for  the  determination  of  TJ!/,  and  the  value  of  e  computed  from 
e  =  sec  4  —  1/1  +  tan2  4 

should  agree  with  that  derived  from  equation  (88).     When  e  differs 
but   little  from  unity,  it  is  conveniently  and  accurately  computed 

from 

e  =  1  +  2  sin2  ^  sec  4. 

The  value  of  a  may  be  found  from 

'  •  (97; 


264  THEORETICAL   ASTRONOMY. 

or  from 


16s'2  rr"  cos2  £  (u"  —  u)  (a/2  —  x'J 

which  is  derived  directly  from  (89),  observing  that  the  elliptic  semi- 
transverse  axis  becomes  negative  in  the  case  of  the  hyperbola. 

As  soon  as  to  has  been  found,  we  derive  from  u,  uf,  and  uff  the 
corresponding  values  of  v,  vf,  and  v",  and  then  compute  the  valuos 
of  F}  F'y  and  F"  by  means  of  the  formula  (57)x ;  after  which,  by 
means  of  the  equation  (69)D  the  corresponding  values  of  N,  Nf,  and 
N'r  will  be  obtained.  Finally,  the  time  of  perihelion  passage  w'll 
be  given  by 

A0A/  0  0 

wherein  log^^  =  7.87336575. 

The  cases  of  hyperbolic  orbits  are  rare,  and  in  most  of  those  which 
do  occur  the  eccentricity  will  not  differ  much  from  that  of  the  para- 
bola, so  that  the  most  accurate  determination  of  T  will  be  effected  by 
means  of  Tables  IX.  and  X.  as  already  illustrated. 

93.  EXAMPLE. — To  illustrate  the  application  of  the  principal  for- 
mula which  have  been  derived  in  this  chapter,  let  us  take  the  follow- 
ing observations  of  Eurynome  ©  : 

Ann  Arbor  M.  T.  (rtfo,  @)u 

1863  Sept.  14  15*  53W  37'.2        1*    0"  44-.91        +  9°  53'  30".8, 
21     9   46    18 .0        0   57      3 .57  9    13    5  .5, 

28     8  49    29 .2        0  52    18 .90        +  8    22     8  .7. 

The  apparent  obliquity  of  the  ecliptic  for  these  dates  was,  uespect- 
*  ively,  23°  27'  20".75,  23°  27'  20".71,  and  23°  27'  20".65 ;  and,  by 
means  of  these,  converting  the  observed  right  ascensions  and  declina- 
tions into  apparent  longitudes  and  latitudes,  we  get — 

Ann  Arbor  M.  T.                           Longitude.  Latitude. 

1863  Sept.  14  15*  53-  37'.2  17°  47'  37".60  -f  3°    8'  43".19, 

21     9   46    18 .0  16    41  36  .20  2    52  27  .46, 

28     8   49    29 .2  15    16  56  .35  +2    32  42  .98. 

For  the  same  dates  we  obtain  from  the  American  Nautical  Alma 
the  following  places  of  the  sun : 


NUMERICAL   EXAMPLE.  265 

True  Longitude.  Latitude.  log  .R0. 

172°    l'42".l  —0.07  0.0022140, 

178    37  17  .2  +  0.77  0.0013857, 

185    26  54  .8  +  0.67  0.0005174. 

Since  the  elements  are  supposed  to  be  wholly  unknown,  the  places 
of  the  planet  must  be  corrected  for  the  aberration  of  the  fixed  stars 
as  given  by  equations  (1).  Thus  we  find  for  the  corrections  to  be 
applied  to  the  longitudes,  respectively, 

-  18".48,  -  19".49,  —  20".8, 

and  for  the  latitudes, 

+  0".47,  +  0".30,  +  0".14. 

When  these  corrections  are  applied,  we  obtain  the  true  places  of  the 
planet  for  the  instants  when  the  light  was  emitted,  but  as  seen  from 
the  places  of  the  earth  at  the  instants  of  observation. 

Next,  each  place  of  the  sun  must  be  reduced  from  the  centre  of 
the  earth  to  the  point  in  which  a  line  drawn  from  the  planet  through 
the  place  of  the  observer  cuts  the  plane  of  the  ecliptic.  For  this 
purpose  we  have,  for  Ann  Arbor, 

<?'  =  42°  5'.4,  log  ft,  =  9.99935  ; 

and  the  mean  time  of  observation  being  converted  into  sidereal  time 
gives,  for  the  three  observations, 

00  ==  3*  29™  1-,  el  =  21*  48-  17',  00"  =  21*  18-  55', 

which  are  the  right  ascensions  of  the  geocentric  zenith,  of  which  y>' 
is  in  each  case  the  declination.  From  these  we  derive  the  longitude 
and  latitude  of  the  zenith  for  each  observation,  namely, 

4,=      60°33'.9,  4'=    350°35'.2,  ^'  =    342°  59'.2, 

60  =  +  22    25.0,  &0'=4-50    50.9,  &0"=  +  53    41.6. 

Then,  by  means  of  equations  (4),  we  obtain 


0  =  —  18".92,  A©'  =  —  36".94,  A0"  =  —  25".76, 

A  log  EQ  =  —  0.0001084,  A  log  Rj  =  —  0.0002201, 

A  log  #0"  =  —  0.0002796. 

For  the  reduction  of  time,  we  have  the  values  +  OM5,  -j~  0*.28,  and 
-f-  08.34,  which  are  so  small  that  they  may  be  neglected. 


2G6  THEOEETICAL   ASTRONOMY. 

Finally,  the  longitudes  of  both  the  sun  and  planet  are  reduced  to 
the  mean  equinox  of  1863.0  by  applying  the  corrections 

—  50".95,  -51".  52,  —  52".14; 

and  the  latitudes  of  the  planet  are  reduced  to  the  ecliptic  of  the  same 
date  by  applying  the  corrections  —  Or/.15,  —  Or/.14,  and  —  Or/.14, 
respectively. 

Collecting  together  and  applying  the  several  corrections  thus  ob- 
tained for  the  places  of  the  sun  and  of  the  planet,  reducing  the  un- 
corrected  times  of  observation  to  the  meridian  of  Washington,  and 
expressing  them  in  days  from  the  beginning  of  the  year,  we  have  the 
following  data  :  — 

tQ  =  257.68079,          A  =  17°  46'  28".17,          0  =  +  3°    8'  43".51, 

tj  =  264.42570,          A'  =  16    40  25  .19,          f  =      2    52  27  .62, 

C'  =  271.38625,          A"  =  15    1544.03,          P"  =  +  2    3242.98, 

O    =172°    0'32".23,  log£  =0.0021056, 

O'  =178    35  48  .74,  logjR'  =0.0011656, 

0"=185    25  36  .90,  log  R"  =  0.0002378. 

The  numerical  values  of  the  several  corrections  to  be  applied  to 
the  data  furnished  by  observation  and  by  the  solar  tables  should  be 
checked  by  duplicate  calculation,  since  an  error  in  any  of  these  re- 
ductions will  not  be  indicated  until  after  the  entire  calculation  of  the 
elements  has  been  effected. 

By  means  of  the  equations 


m(0"  —  0Q  _  EEfsm(Q'—  Q) 

"  RR"  sin  (O"  —  O)  '  RR"  sin  (©"—  0)' 

tan/3'  tan  (A'  -Q') 


cosrf        ' 

we  obtain 

log  N=  9.7087449,  log  N"  =  9.6950091, 

4/  =  161°42'13".16, 
log  (R'  sin  V)  =  9.4980010,  log  (R  cos  4/)  =  9.9786355.. 

The  quadrant  in  which  -\}/  must  be  taken  is  determined  by  the  con- 
ditions that  i//  must  be  less  than  180°,  and  that  cos^'  and  cos  (A'—  O') 
must  have  the  same  sign.  Then  from 


NUMEKICAL   EXAMPLE. 

tan  Jsin  Q  (/'  +  A)  -  K)  =  |^gf  ,  sec  J  (/'  -  i), 
tanlcosQ  (A"  +  A)  -  K)  =  "  ,  cosec  |      - 


267 


—  JST) 


G"—  JjQ  «  sec/3'  jfcft"  sm(G"—  0) 

0  J~  sin  (A"—  A)'  a0  sin  (/>/'  —  A)~~ 

we  compute  K,  I,  $,,  a0,  6,  c,  c?,  /,  and  h.  The  angle  J  must  be  less 
than  90°,  and  the  value  of  ft0  must  be  determined  with  the  greatest 
possible  accuracy,  since  on  this  the  accuracy  of  the  resulting  elements 
principally  depends.  Thus  we  obtain 

K=  4°  47'  29".48,  log  tan  1=  9.3884640, 

A  —  2°  52'  59"f  if  ,  log  a0  =  6.8013583n, 

log  6  =  2.5456342n,  log  c  =  2.2328550W, 

log  d  =  1.2437914,          log/=  1.3587437n,         log  h  =  3.924769L 

The  formulae 


,  _  sin  (A;  —  /I)          ^  sin  (/I  —  Q) 

"  J  " 


gve 


sin  (A"  —  A) 
sin  (/"  —  K) 


log  M,  =  9.8946712, 
log  Jfa  =  1.9404111, 


hsm(l  — 


log  Mf  =  9.6690383, 
log  Mz"  =  0.7306625n. 


The  quantities  thus  far  obtained  remain  unchanged  in  the  suc- 
cessive approximations  to  the  values  of  P  and  Q. 
For  the  first  hypothesis,  from 


i?0  sin  Z,=R  sin  4/, 

T)Q  cos  C  =  &0  —  -R'  cos  4*', 


268  THEORETICAL   ASTRONOMY. 

we  obtain 

log  r  =  9.0782249,  log  T"  =  9.0645575, 

log  P  =  9.9863326,  log  Q  =  8.1427824, 

log  c0  ==  2.2298567n,  log  k0  =  0.0704470, 

log  1Q  =  0.0716091,  log  7y0  =  0.3326925, 

C  =  8°  24'  49".74,  Iogm0  ==  1.2449136. 

The  quadrant  in  which  £  must  be  situated  is  determined  by  the  con- 
dition that  ^0  shall  have  the  same  sign  as  b 

The  value  of  z1  must  now  be  found  by  trial  from  the  equation 

sin  (si  —  C)  =  m0  sin4  sf. 

Table  XII.  shows  that  of  the  four  roots  of  this  equation  one  exceeds 
180°,  and  is  therefore  excluded  by  the  condition  that  sin/  must  be 
positive,  and  that  two  of  these  roots  give  zf  greater  than  180°  —  ij/, 
and  are  excluded  by  the  condition  that  zr  must  be  less  than  180°—  -J/. 
The  remaining  root  is  that  which  belongs  to  the  orbit  of  the  planet, 
and  it  is  shown  to  be  approximately  10°  40'  ;  but  the  correct  value 
is  found  from  the  last  equation  by  a  few  trials  to  be 

z'  =  9°  1'22".96. 

The  root  which  corresponds  to  the  orbit  of  the  earth  is  18°  20'  41", 
and  differs  very  little  from  180°  —  ty. 
Next,  from 


we  derive 

log  r'  =  0.3025672,  log  P'  =  0.0123991, 

''      log  n  =  9.7061229,  log  n"  =  9.6924555, 

log  p  =  0.0254823,  log  f  =  0.0028859. 

The  values  of  the  curtate  distances  having  thus  been  found,  the 
heliocentric  places  for  the  three  observations  are  now  computed  from 


NUMERICAL   EXAMPLE.  269 

r  cos  b  cos  (I  —  O)        =p  cos  (A  —  Q)  —  R, 

r  cos  b  sin  (I  —  Q)         =^sin(A  —  Q), 

r  sin  &  =  f>  tan  /?  ; 

/  cos  6'  cos  (V  —  Q  0     =y  cos  (A'  —  ©')  —  #, 

/  cos  V  sin  (r  —  .0')      =  Pf  sin  (A'  —  0  '), 

r'  sin  b'  —  />'  tan  p  ; 

r"  cos  6"  cos  (r  —  Q")  =  P"  cos  (A"  —  0")  —  J2", 

r"  cos  6"  sin  (f  —  0")  =  /e,"  sin  (A"  —  0"), 


which  give 

J  =  5°  14'  39".53,  log  tan  b  =8.4615572,  logr  =0.3040994, 
V  =  7  45  11  .28,  log  tan  b'  =8.4107555,  logr'  =0.3025673, 
I"  =  10  21  34  .57,  log  tan  b"  =  8.3497911,  log  r"  =  0.3011010. 

The  agreement  of  the  value  of  logr'  thus  obtained  with  that  already 
found,  is  a  proof  of  part  of  the  calculation.  Then,  from 

(\  nn  .    7^        r^\        tan  6"-}-  tan  6 
nG(/  +  0  -  Q)  =          .  . 


cos  i  cos  ^  cos  i 

we  get 

ft  =  207°  2'  38".16,  t  =  4°  27'  23".84, 

u  =  158°  8'  25".78,          uf  =  160°  39'  18".13,          u"  =  163°  16'  4".42. 

The  equation 

tan  b'  =  tan  i  sin  (T  —  ft  ) 

gives  log  tan  br  =  8.4107514,  which  differs  0.0000041  from  the  value 
already  found  directly  from  //.  This  difference,  however,  amounts 
to  only  Or/.05  in  the  value  of  .the  heliocentric  latitude,  and  is  due  to 
errors  of  calculation.  If  we  compute  n  and  n"  from  the  equations 

r'r"  sin  (u"—  u')  rrr  sin  (u'  —  u) 


n  =     „   .    ;  „ ±,  n"  = 


rr"  sin  (u"  —U*)'  ~  rr"  sin  («"  —  u) ' 

the  results  should  agree  with  the  values  of  these  quantities  previously 
computed  directly  from  P  and  Q.  Using  the  values  of  u,  u',  and 
u"  just  found,  we  obtain 

log  n  =  9.7061158,  log  n"  =  9.6924683, 


270  THEORETICAL   ASTRONOMY. 

which  differ  in  the  last  decimal  places  from  the  values  used  in  finding 
p  and  pff.  According  to  the  equations 

d  log  n  =  —  21.055  cot  (u"—  «')  du', 
d  log  n"  =  -f  21.055  cot  (ur  —  u)  du', 

the  differences  of  logn  and  logn"  being  expressed  in  units  of  the 
seventh  decimal  place,  the  correction  to  u'  necessary  to  make  the  two 
values  of  logn  agree  is  — 0".15;  but  for  the  agreement  of  the  two 
values  of  log??/',  u'  must  be  diminished  by  0".26,  so  that  it  appears 
that  this  proof  is  not  complete,  although  near  enough  for  the  first 
approximation.  It  should  be  observed,  however,  that  a  great  circle 
passing  through  the  extreme  observed  places  of  the  planet  passes 
very  nearly  through  the  third  place  of  the  sun,  and  hence  the  values 
of  p  and  p"  as  determined  by  means  of  the  last  two  of  equations  (18) 
are  somewhat  uncertain.  In  this  case  it  would  be  advisable  to  com- 
pute p  and  p",  as  soon  as  pf  has  been  found,  by  means  of  the  equa- 
tions (22)  and  (23).  Thus,  from  these  equations  we  obtain 

log  p  =  0.025491 8,  log  p"  =  0.0028874, 

and  hence 

I  =  5°14'40".05,  log  tan  b  =8.4615619,  log  r  =  0.3041042, 
J"=10  2134.19,  log  tan  b"  =8.3497919,  log/' =0.3011017, 

&  ==  207°  2'  32".97,  i  =  4°  27'  25".13, 

u  =  158°  8'  31".47,          u'  =  160°  39'  23".31,          u"  =  163°  16'  9".22. 

The  value  of  log  tan  bf  derived  from  Xf  and  these  values  of  Q,  and  i, 
is  8.4107555,  agreeing  exactly  with  that  derived  from  pf  directly. 
The  values  of  n  and  n"  given  by  these  last  results  for  u,  u'  and  u"y 
are 

log  n  =  9.7061144,  log  n"  =  9.6924640 ; 

and  this  proof  will  be  complete  if  we  apply  the  correction  duf= — 0".18 
to  the  value  of  u',  so  that  we  have 

u"  —u'  =  2°  36'  46".09,  u'  —  u  =  2°  30'  51".66. 

The  results  which  have  thus  been  obtained  enable  us  to  proceed  to 
a  second  approximation  to  the  correct  values  of  P  and  §,  and  we 
may  also  correct  the  times  of  observation  for  the  time  of  aberration 
by  means  of  the  formulae 

t  =  tQ—Cp  sec  /?,  t  =  tQ'  —  Cp'  sec  p,  t"  =  t0"  —  Cp"  sec  p', 

wherein  log  C=  7.760523,  expressed  in  parts  of  a  day.   Thus  we  get 

t  =  257.67467,  if  =  264.41976,  *"  =  271.38044, 


NUMERICAL   EXAMPLE.  271 

and  hence 

log  r  =  9.0782331,  log  r'  =  9.3724848,  log  T"  =  9.0645692. 

Then,  to  find  the  ratios  denoted  by  s  and  s",  we  have 


sin  Y  cos  Q  =  sin  |  (u"  —  t/)> 

sin  Y  sin  G  =  cos  J  (u"  —  u')  cos  2/, 

cos  7-  =  cos  ^  (u"  —  u')  sin  2/  ; 


tan/'  = 

sin  f'  cos  G"  =  sin  ^  (V  —  w), 

sin  r"  sin  G"  =  cos  J  (V  —  t*)  cos  2/', 

cos  /'  =  cos  £  (M'  —  w)  sin  2/' ; 

r2  .  sin2 

")3cos3/ 


from  which  we  obtain 

y  =  44°  57f    6".00,  /'  =  44°  56f  57".50, 

r=   1    18  35  .90,  r"=   1    15  40  .09, 

logm  =  6.3482114,  logm';  =  6.3163548, 

log.;  =  6.1163135,  log/'  =  6.0834230. 

From  these,  by  means  of  the  equations 

m  m 




V      


using  Tables  XIII.  and  XIV.,  we  compute  s  and  s".     First,  in  the 
case  of  s,  we  assume 

y  =  _       =  0.0002675, 


and,  with  this  as  the  argument,  Table  XIII.  gives  log  s2  =  0.0002581  . 
Hence  we  obtain  xf  =  0.000092,  and,  with  this  as  the  argument, 
Table  XIV.  gives  c  =  0.00000001  ;  and,  therefore,  it  appears  that  a 
repetition  of  the  calculation  is  unnecessary.  Thus  we  obtain 

logs  =0.0001290,  logs"  =0.0001  200. 

When  the  intervals  are  small,  it  is  not  necessary  to  use  the  formulae 


272  THEORETICAL   ASTRONOMY. 

in  the  complete  form  here  given,  since  these  ratios  may  then  be  found 
by  a  simpler  process,  as  will  appear  in  the  sequel.     Then,  from 


TT"  r" 

ss"    rr"  cos  J  (it"  —  w')  cos  J  (it"  —  tt)  cos  ^  (M'  —  «)' 
we  find 

log  P  =  9.9863451,  log  Q  =  8.1431341, 


with  which  the  second  approximation  may  be  completed.  We  now 
compute  c0,  7cQ,  10,  z' ',  &c.  precisely  as  in  the  first  approximation ;  but 
we  shall  prefer,  for  the  reason  already  stated,  the  values  of  p  and  p" 
computed  by  means  of  the  equations  (22)  and  (23)  instead  of  those 
obtained  from  the  last  two  of  the  formulae  (18).  The  results  thus 
derived  are  as  follows : — 

log  c0  =  2.2298499n,  log  Jc0  =  0.0714280, 

log  10  =  0.0719540,  log  i?0  =  0.3332233, 

C  =  8°  24'  12".48,  log  m0  =  1.2447277, 

zf  =  9°  0'  30".84, 

log  /  —  0.3032587,  log  p'  =  0.0137621, 

log  n  ==  9.7061153,  log  n"=  9.6924604, 

log/>  =  0.0269143,  log/'  =  0.0041748, 

I   =   5M5'57".26,        log  tan  b  =8.4622524,         logr  =0.3048368, 

I'  =   7    46     2.76,        log  tan  V  =  8.4114276,         log/  =  0.3032587, 

lrf  =  10    22     0  .91,        log  tan  b"  =  8.3504332,         log  r"  =  0.3017481, 

a  =  207°  0'  0".72,  *  =  4°  28r  35'r.20, 

u  =  158°  12'  19".54,         u'  =  160°  42'  45".82,         u"  =  163°  19'  7".14. 

The  agreement  of  the  two  values  of  logr'  is  complete,  and  the  value 
of  log  tan  b'  computed  from 

tan  b'  =  tan  i  sin  (lr  —  Q> ), 

is  log  tan  bf  =  8. 41 1427 9,  agreeing  with  the  result  derived  directly 
from  pr.  The  values  of  n  and  n"  obtained  from  the  equations  (54) 

are 

log  n  =  9.7061156,  log  n"  =  9.6924603, 

which  agree  with  the  values  already  used  in  computing  p  and  p",  and 
the  proof  of  the  calculation  is  complete.  We  have,  therefore, 

tt"_  u>  =  2°  36'  21".32,    u'—  u  =  2°  30'  26".28,    u"—  u  =  5°&  47".60. 

From  these  values  of  iin  —  ur  and  uf  —  u,  we  obtain 
log  s  =  0.0001284,  logs"  =  0.0001193, 


NUMERICAL   EXAMPLE.  273 

and,  recomputing  P  and  Q,  we  get 

log  P  ==  9.9863452,  log  Q  =  8.1431359, 

which  differ  so  little  from  the  preceding  values  of  these  quantities 
that  another  approximation  is  unnecessary.    We  may,  therefore,  from 
the  results  already  derived,  complete  the  determination  of  the  elements 
of  the  orbit. 
The  equations 


sin  •/  cos  G'  =  sin  -^  (u"  —  u\ 
sin  /  sin  Gr  =  cos  J  (u"  —  u)  cos  2/, 
cos  /  =  cos  £  (u"  —  Uj  sin  2/, 
m,=  v_sm 

"3  J  "" 


(r-j-r")3cosV 
give 

/  =  44°  53'  53".25,         /  =  2°  33'  52".97,         log  tan  G'  =  8.9011435 
log  m'  =  6.9332999,  log/  =  6.7001345. 

From  these,  by  means  of  the  formulae 

'_      m'  '—  —  —  •• 

""!+/+«*  ~*'a   '^' 

and  Tables  XIII.  and  XIV.,  we  obtain 

logs'2<=:  0.0009908,  loga/  =  6.549411U 

Then  from 

we  get 

logjs  =  0.3691818. 

The  values  of  logp  given  by 

s/r"  sin  (i*"  —  w')  \2      /  s'Vr'  sin  (wf  —  u)  \" 


are  0.3691824  and  0.3691814,  the  mean  of  which  agrees  with  the 
result  obtained  from  u"  —  u,  and  the  differences  between  the  separate 
results  are  so  small  that  the  approximation  to  P  and  Q  is  sufficient, 
The  equations 


P 

cos  <p  =  —  - 


18 


274  THEORETICAL   ASTRONOMY. 

give 


"  —  E)  =  1°  4'  42".903,  log  (a  cos  ?)  =  0.3770315, 

log  cos  <p  =  9.9921503. 
Next,  from 

e  sin  (at  —  I  (n"  +  u))  =  -  P  /  _  tan  G', 
cos  /  V  rr" 

e  cos  («,  -  J  K  +  u))  =  -  ^—  -  seci  (u"  -  v), 
cos/yrr 

we  obtain 

a>  =  190°  15'  39".57,  log  e  =  log  sin  ?  =  9.2751434, 

?>  ==    10    51  39  .62,  TT  =  «,  -f-  £  =  37°  15'  40".29. 

This  value  of  ^  gives  log  cos<p  =  9.9921501,  agreeing  with  the  result 
already  found. 

To  find  a  and  //,  we  have 

p  k 


the  value  of  k  expressed  in  seconds  of  arc  being  log  k  =  3.5500066, 
from  which  the  remits  are 

log  a  =  0.3848816,  log  ft  =  2.9726842. 

The  true  anomalies  are  given  by 

v  =  u  —  at,  vf  =  u'  —  o>,  v"  =  u"  —  cw, 

according  to  which  we  have 
v  =  327°  56'  39".97,  v'  =  330°  27'  6".25,          i"  =  333°  3'  27".57. 

If  we  compute  r,  rf,  and  r"  from  these  values  by  means  of  the  polar 
equation  of  the  ellipse,  we  get 

log  r  =  0.3048367,  log  r'  =  0.3032586,  log  r"  =  0.3017481, 

and  the  agreement  of  these  results  with  those  derived  directly  from 
p,  pf,  and  prr  is  a  further  proof  of  the  calculation. 
The  equations 

tan  {  E  =  tan  (45°  —  £?)  tan  £v, 

tan  ^E'  =  tan  (45°  —  1$0  tan  ^, 

tan  JJE"  =  tan  (45°  —  £?)  tan  W 
give 

E  =  333°  17'  28'  .18,       E'  =  335°  24'  38".00,      E"  =  337°  36'  19".78. 


NUMERICAL   EXAMPLE.  275 

The  \alue  of  J  (E"  —  E)  thus  obtained  differs  only  0".003  from  that 
computed  directly  from  x'  '. 

Finally,  for  the  mean  anomalies  we  have 

M  =  E  —  e  sin  E,          M'  =  E'  —  e  sin  E'y         M"  =  E"  —  e  sin  E'\ 
from  which  we  get 
M  •=  338°  8'  36".71,       M'  =  339°  54'  10".61,       M"  =  341°  43'  6".97  ; 

and  if  Jf0  denotes  the  mean  anomaly  for  the  date  T=1863  Sept.  21.5 
Washington  mean  time,  from  the  formulae 

M=M  —n(t  —  r> 


we  obtain  the  three  values  339°  55'  25".97,  339°  55'  25".96,  and 
339°  55'  25".96,  the  mean  of  which  gives 

M9  =  339°  55'  25".96. 

The  agreement  of  the  three  results  for  Jf0  is  a  final  proof  of  the 
accuracy  of  the  entire  calculation  of  the  elements. 

Collecting  together  the  separate  results  obtained,  we  have  the  fol- 
lowing elements  : 

Epoch  =  1863  Sept.  21.5  Washington  mean  time. 
M  =  339°  55'  25".96 
*=   37    15  40  .29) 
ft  -  207      0     0   72  V  EchPtlc  and  Mean 

06   —  •*J"  '          v       v    •  I  *i   /         .„        .  -<o/>or\ 

t=     4    28  35.20J      Equinox  1863.0. 
<P  =   10    51  39  .62 
'      log  a  =  0.384881  6 
log  ^  =  2.9726842 
At  =  939".04022. 

If  we  compute  the  geocentric  right  ascension  and  declination  of 
the  planet  directly  from  these  elements  for  the  dates  of  the  observa- 
tions, as  corrected  for  the  time  of  aberration,  and  then  reduce  the 
observations  to  the  centre  of  the  earth  by  applying  the  corrections 
for  parallax,  the  comparison  of  the  results  thus  obtained  will  show 
how  closely  the  elements  represent  the  places  on  which  they  are 
based.  Thus,  we  compute  first  the  auxiliary  constants  for  the  equator, 
using  the  mean  obliquity  of  the  ecliptic, 

e  =  23°  27'  24".96, 


276  THEORETICAL   ASTRONOMY. 

and  the  following  expressions  for  the  heliocentric  co-ordinates  of  the 
planet  are  obtained  : 

x  =  r  [9.9997272]  sin  (296°  55'  46".05  -f  u), 
y=r  [9.9744699]  sin (206  12  42  .79  +  u), 
z  =  r  [9.5249539]  sin  (212  39  14  .62  +  u). 

The  numbers  enclosed  in  the  brackets  are  the  logarithms  of  sin  a, 
sin bj  and  sine,  respectively;  and  these  equations  give  the  co-ordinates 
referred  to  the  mean  equinox  and  equator  of  1863.0. 

The  places  of  the  sun  for  the  corrected  times  of  observation,  and 
referred  to  the  mean  equinox  of  1863.0,  are 


True  Longitude. 

Latitude. 

Log£. 

172°    0'  29".5 

—  0".07 

0.0022146, 

178    36    4  .5 

+  0  .77 

0.0013864, 

185    25  42  .0 

+  0  .67 

0.0005182. 

If  we  compute  from  these  values,  by  means  of  the  equations  (104),, 
the  co-ordinates  of  the  sun,  and  combine  them  with  the  corresponding 
heliocentric  co-ordinates  of  the  planet,  we  obtain  the  following  geo- 
centric places  of  the  planet : 

a   =  15°  10'  29".06,  8  =  -f  9°  53'  16".72,  log  J   —  0.02726, 

a'  =  14    15     0  .22,  3'  =      912  51  .29,  log  A'  =  0.01410, 

a"  =  13      3  49  .47,  *"  =  +  8    21  54  .46,  log  A"  =  0.00433. 

To  reduce  these  places  to  the  apparent  equinox  of  the  date  of  obser- 
vation, the  corrections 

-f  48".14,  -f-  48".54,  +  48".91, 

must  be  applied  to  the  right  ascensions,  respectively,  and 
-f  18".55,  +  18".92,  +  19".31, 

to  the  declinations.     Thus  we  obtain  : 

Washington  M.  T.  Comp.  a.  Comp.  d. 

1863  Sept.  14.67467  1»    0™  45M5  +  9°  53'  35".3, 

21.41976  0   57      3 .25  9    13  10  .2, 

28.38044  0   52    18.  56  -f  8    22  13  .8. 

The  corrections  to  be  applied  to  the  respective  observations,  in  order 
to  reduce  them  to  the  centre  of  the  earth,  are  -f  0*.24,  —  0*.31,  —  08.34 
in  right  ascension,  and  -f  4".5,  +  4".8,  -f  5".l  in  declination,  so 
that  we  have,  for  the  same  dates, 


NUMERICAL   EXAMPLE.  277 

Observed  a.  Observed  <5. 

lh    Qm  45M5  +  9°  53'  35".3, 

0   57      3 .26  9    13  10  .3, 

0  52    18 .56  +8    22  13  .8. 

The  comparison  of  these  with  the  computed  values  shows  that  the 
extreme  places  are  exactly  represented,  while  the  difference  in  the 
middle  place  amounts  to  only  0*.01  in  right  ascension,  and  to  0".l 
in  declination.  It  appears,  therefore,  that  the  observations  are  com- 
pletely satisfied  by  the  elements  obtained,  and  that  the  preliminary 
corrections  for  aberration  and  parallax,  as  determined  by  the  equa- 
tions (1)  and  (4),  have  been  correctly  computed. 

It  cannot  be  expected  that  a  system  of  elements  derived  from  ob- 
servations including  an  interval  of  only  fourteen  days,  will  be  so 
exact  as  the  results  which  are  obtained  from  a  series  of  observations 
or  from  those  including  a  much  longer  interval  of  time;  and  although 
the  elements  which  have  been  derived  completely  represent  the  data, 
yet,  on  account  of  the  smallness  of  /9r  —  /90,  this  difference  being  only 
31  ".893,  the  slight  errors  of  observation  have  considerable  influence 
in  the  final  results. 

When  approximate  elements  are  already  known,  so  that  the  cor- 
rection for  parallax  may  be  applied  directly  to  the  observations,  in 
order  to  take  into  account  the  latitude  of  the  sun,  the  observed  places 
of  the  body  must  be  reduced,  by  means  of  equation  (6),  to  the  point 
in  which  a  perpendicular  let  fall  from  the  centre  of  the  earth  to  the 
plane  of  the  ecliptic  cuts  that  plane.  The  times  of  observation  must 
also  be  corrected  for  the  time  of  aberration,  and  the  corresponding 
places  of  both  the  planet  and  the  sun  must  be  reduced  to  the  ecliptic 
and  mean  equinox  of  a  fixed  epoch;  and  further,  the  reduction  to 
the  fixed  ecliptic  should  precede  the  application  of  equation  (6). 

If  the  intervals  between  the  times  of  observation  are  considerable, 
it  may  become  necessary  to  make  three  or  more  approximations  to  the 
values  of  P  and  §,  and  in  this  case  the  equations  (82)  may  be  applied. 
But  when  approximate  elements  are  already  known,  it  will  be  advan- 
tageous to  compute  the  first  assumed  values  of  P  and  Q  directly 
from  these  elements  by  means  of  the  equations  (44)  or  by  means  of 
(48)  and  (51) ;  and  the  ratios  s  and  s"  may  be  found  directly  from  the 
equations  (46).  In  the  case  of  very  eccentric  orbits  this  is  indispen- 
sable, if  it  be  desired  to  avoid  prolixity  in  the  numerical  calculation, 
since  otherwise  the  successive  approximations  to  P  and  Q  will  slowly 
approach  the  limits  required. 


278  THEORETICAL   ASTRONOMY. 

The  various  modifications  of  the  formulae  for  certain  special  cases, 
as  well  as  the  formulae  which  must  be  used  in  the  case  of  parabolic 
and  hyperbolic  orbits,  and  of  those  differing  but  little  from  the 
parabola,  have  been  given  in  a  form  such  that  they  require  no  fur- 
ther illustration. 

94.  In  the  determination  of  an  unknown  orbit,  if  thje  intervals  are 
considerably  unequal,  it  will  be  advantageous  to  correct  the  first 
assumed  value  of  P  before  completing  the  first  approximation  in  the 
manner  already  illustrated.  The  assumption  of 


is  correct  to  terms  of  the  fourth  order  with  respect  to  the  time,  and 
for  the  same  degree  of  approximation  to  P  we  must,  according  to 
equation  (28)3,  use  the  expression 

p  — L 


T 

which  becomes  equal  to  —  only  when  the  intervals  are  equal.     The 
first  assumed  values 


furnish,  with  very  little  labor,  an  approximate  value  of  rf ;  and  then, 
with  the  values  of  P  and  §,  derived  from 


T"    i 
=V 


(98) 


the  entire  calculation  should  be  completed  precisely  as  in  the  example 
given.  Thus,  in  this  example,  the  first  assumed  values  give 

log  r'  =  0.30257, 

and,  recomputing  P  by  means  of  the  first  of  these  equations,  we  get 
log  P  =  9.9863404,  log  Q  =  8.1427822, 

with  which,  if  the  first  approximation  to  the  elements  be  completed, 
the  results  will  differ  but  little  from  those  obtained,  without  this  cor- 
rection, from  the  second  hypothesis.  If  the  times  had  been  already 
corrected  for  the  time  of  aberration,  the  agreement  would  be  still 
closer. 

The  comparison  of  equations  (46)  with  (25)s  gives,  to  terms  of  the 
fourth  order, 


NUMERICAL   EXAMPLE.  279 


and,  if  the  intervals  are  equal,  this  value  of  s'  is  correct  to  terms  of 
the  fifth  order.     Since 

log.a  =  log.(l  +  («-!))=«-!  -£(«-!)«  +  Ac., 
we  have,  neglecting  terms  of  the  fourth  order, 

l°g»  =  £-pl>  (99) 

in  which  log  JA0=  8.8596330.     We  have,  also,  to  the  same  degree  of 
approximation, 

!<«  '=£•71'  to^  =  £-5>  (100) 

For  the  values 

log  r  =  9.0782331,  log  rf  =  9.3724848,  log  r"  **  9.0645692, 

log/  =  0.3032587, 
these  formulae  give 

log  s  =  0.0001277,  log  s'  =  0.0004953,  log  «"  =  0.0001199, 

which  differ  but  little  from  the  correct  values  0.0001284,  0.0004954, 
and  0.0001193  previously  obtained. 
Since 

sec8  /  =  1  +  6  sin2  tf  +  &c., 


the  second  of  equations  (65)  gives 


sin'      &c- 


Substituting  this  value  in  the  first  of  equations  (66),  we  get 


If  we  neglect  terms  of  the  fourth  order  with  respect  to  the  time,  it 
will  be  sufficient  in  this  equation  to  put  yr  =  f,  according  to  (71),  and 
hence  we  have 


4 

3 


and,  since  sf  —  1  is  of  the  second  order  with  respect  to  r',  we  have, 
to  terms  of  the  fourth  order, 


280  THEORETICAL   ASTRONOMY. 

Therefore, 


which,  when  the  intervals  are  small,  may  be  used  to  find  sf  from  r 
and  r".     In  the  same  manner,  we  obtain 

l^r,-        (102) 


For  logarithmic  calculation,  when  addition  and  subtraction  loga- 
rithms are  not  used,  it  is  more  convenient  to  introduce  the  auxiliary 
angles  £,  £r,  and  £",  by  means  of  which  these  formulae  become 

KtW  (103) 


in  which  log  |^0=  9.7627230.     For  the   first   approximation  these 

equations  will  be  sufficient,  even  when  the  intervals  are  considerable, 

to  determine  the  values  of  s  and  s"  required  in  correcting  P  and  Q. 

The  values  of  r,  r',  r",  and  r"  above  given,  in  connection  with 

log  r  =  0.3048368,  log  r"  =  0.3017481, 

give 

log  5  =  0.0001284,  log  sf  =  0.0004951,  log  s"  =  0.0001193. 

These  results  for  logs  and  logs"  are  correct,  and  that  for  logs'  differs 
only  3  in  the  seventh  decimal  place  from  the  correct  value. 


ORBIT   FEOM   FOUK   OBSERVATIONS. 


CHAPTER  V. 

DETERMINATION   OF   THE   ORBIT   OF  A  HEAVENLY  BODY  FROM   FOUR  OBSERVATIONS, 
OF   WHICH   THE   SECOND   AND   THIRD   MUST  BE   COMPLETE. 

95.  THE  formulae  given  in  the  preceding  chapter  are  not  sufficient 
to  determine  the  elements  of  the  orbit  of  a  heavenly  body  when  its 
apparent  path  is  in  the  plane  of  the  ecliptic.  In  this  case,  however, 
the  position  of  the  plane  of  the  orbit  being  known,  only  four  ele- 
ments remain  to  be  determined,  and  four  observed  longitudes  will 
furnish  the  necessary  equations.  There  is  no  instance  of  an  orbit 
whose  inclination  is  zero  ;  but,  although  no  such  case  may  occur,  it  may 
happen  that  the  inclination  is  very  small,  and  that  the  elements 
derived  from  three  observations  will  on  this  account  be  uncertain, 
and  especially  so,  if  the  observations  are  not  very  exact.  The  diffi- 
culty thus  encountered  may  be  remedied  by  using  for  the  data  in  the 
determination  of  the  elements  one  or  more  additional  observations, 
and  neglecting  those  latitudes  which  are  regarded  as  most  uncertain. 
The  formulae,  however,  are  most  convenient,  and  lead  most  expe- 
ditiously  to  a  knowledge  of  the  elements  of  an  orbit  wholly  unknown, 
when  they  are  made  to  depend  on  four  observations,  the  second  and 
third  of  which  must  be  complete  ;  but  of  the  extreme  observations 
only  the  longitudes  are  absolutely  required. 

The  preliminary  reductions  to  be  applied  to  the  data  are  derived 
precisely  as  explained  in  the  preceding  chapter,  preparatory  to  a  de- 
termination of  the  elements  of  the  orbit  from  three  observations. 

Let  t,  t',  t",  t'"  be  the  times  of  observation,  r,  r',  r",  r'"  the  radii- 
vectores  of  the  body,  u,  uf,  un  ',  u'"  the  corresponding  arguments  of 
the  latitude,  R,  R',  R",  R"r  the  distances  of  the  earth  from  the  sun, 
and  O,  O',  O",  O'"  the  longitudes  of  the  sun  corresponding  to 
these  times.  Let  us  also  put 


=  r'r'"sm(u'"  —  u'\ 
[rV"]  =  rV"  sin  (u"r  —  u"\ 
and 


282  THEORETICAL   ASTRONOMY. 

Then,  according  to  the  equations  (5)3,  we  shall  have 

nx   —  x'  -f-  ri'x"  =  0, 
ny  _</  +n»y»   =0, 


.Let  ;,  A',  ;/',  A'"  be  the  observed  longitudes,  /9,  /9',  0",  /?'"  the  ob- 
served latitudes  corresponding  to  the  times  £,  tf,  t'r,  t"r,  respectively, 
and  J,  A',  A",  A'"  the  distances  of  the  body  from  the  earth.  Further, 
let 

A'"  cos  f"  =  p'"9 

and  for  the  last  place  we  have 

xf"  =  p'"  cos  A'"  —  K"  cos  O'", 
ym  =  p'"  sin  *'"  —  £"'  sin  O'". 

Introducing  these  values  of  x'"  and  yf",  and  the  corresponding  values 
of  x,  xf,  x",  y,  yf,  y"  into  the  equations  (2),  they  become 

0  =  n  (p  cos  I  —  R  cos  Q  )  —  (>'  cos  A'  —  R'  cos  O') 

+  n"G 

0  =  n  0»  sin  A  —  J2  sin  O  )  —  (f?  sin  Ar  —  R  sin  O' 

+  w" 

0  =  n'  G/  cos  /  -  R  cos  O')  —  (p"  cos  A"  —  #'  cos  O")  (3) 

-f  n'"  (p'"  cos  X'"  —  R"  cos  0'"), 
0  =  w'  (pr  sin  A'  —  ^  sin  O')  —  (p"  sin  X"  —  R"  sin  0") 

4-  n'"  (/>"'  sin  A'"  —  R"  sin  0'"). 

If  we  multiply  the  first  of  these  equations  by  sin  ^,  and  the  second 
by  —  cos  ^,  and  add  the  products,  we  get 

0  =  nR  sin  (A  —  Q)  —  (p*  sin  (X  —  X)  +  Rf  sin  (A  —  0')) 

+  n"  (P"  sin  (A"  —  A)  -f-  #'  sin  (A  —  0  "))  ;  (4) 

and  in  a  similar  manner,  from  the  third  and  fourth  equations,  we 
find 

0  =  n'  (pf  sin  (X"  —  X  )  —  R'  sin  (X"  —  0  '))  (5j) 

-  (p"  sin  (/"—  A")  —  K'  sin  (/"—  ©"))  -  n'"R"  sin  (A'"—  0'"). 

Whenever  the  values  of  n,  n'9  n",  and  n//r  are  known,  or  may  be 
determined  in  functions  of  the  time  so  as  to  satisfy  the  conditions  of 
motion  in  a  conic  section,  these  equations  become  distinct  or  inde- 
Dendent  of  each  other  ;  and,  since  only  two  unknown  quantities  p1 


OEBIT   FROM    FOUR   OBSERVATIONS.  283 

and  pn  are  involved  in  them,  they  will  enable  us  to  determine  these 
curtate  distances. 
Let  us  now  put 

cos  p  sin  (/  —  /I)    =  A,  cos  ft"  sin  (A"  —  X)=B, 

cos  ft"  sin  (/'"—  A")  =  C,  cos  p  sin  (*'"  —  X )  =  D, 

and  the  preceding  equations  give 

Ap'  sec  ^  —  Bn"P"  sec  /5"  =  wJ5  sin  (A  —  0  )  —  Rr  sin  (A  —  Q') 

+  n"J2"Biny  —  0"), 

!>»>'  sec  /?'—  Q/'  sec  /9"=  n'tf  sin  (/"—  0 ')  —  R"  sin  (A'"—  Q")        (7) 

+  n'"R'"  sin  (/"  —  Q'"). 

Tf  we  assume  for  n  and  n"  their  values  in  the  case  of  the  orbit  of 
the  earth,  which  is  equivalent  to  neglecting  terms  of  the  second  order 
in  the  equations  (26)3,  the  second  member  of  the  first  of  these  equa- 
tions reduces  rigorously  to  zero ;  and  in  the  same  manner  it  can  be 
shown  that  when  similar  terms  of  the  second  order  in  the  corre- 
sponding expressions  for  nf  and  n"  are  neglected,  the  second  member 
of  the  last  equation  reduces  to  zero.  Hence  the  second  member  of 
each  of  these  equations  will  generally  differ  from  zero  by  a  quantity 
which  is  of  at  least  the  second  order  with  respect  to  the  intervals  of 
time  between  the  observations.  The  coefficients  of  p'  and  p"  are  of 
the  first  order,  and  it  is  easily  seen  that  if  we  eliminate  p"  from 
these  equations,  the  resulting  equation  for  //  is  such  that  an  error  of 
the  second  order  in  the  values  of  n  and  n"  may  produce  an  error  of 
the  order  zero  in  the  result  for  pf,  so  that  it  will  not  be  even  an 
approximation  to  the  correct  value ;  and  the  same  is  true  in  the  case 
of  prr .  It  is  necessary,  therefore,  to  retain  terms  of  the  second  order 
in  the  first  assumed  values  for  n,  nf,  n",  and  ii'1'  \  and,  since  the 
terms  of  the  second  order  involve  rf  and  rf/,  we  thus  introduce  two 
additional  unknown  quantities.  Hence  two  additional  equations  in- 
volving rf,  r",  p'j  p"  and  quantities  derived  from  observation,  must 
be  obtained,  so  that  by  elimination  the  values  of  the  quantities  sought 
may  be  found. 

From  equation  (34)4  we  have 


P'  sec  p  =  R'  cos  4/  ±  V  r'2  —  Rr*  sin2  4/,  (8) 

which  is  one  of  the  equations  required;  and  similarly  we  find,  for 
the  other  eauation, 


P"  sec  ft"  =  R"  cos  V '  db  y  r'"2—  R"2  sin2  4".  (9) 


284  THEORETICAL   ASTRONOMY. 

Introducing  these  values  into  the  equations  (7),  and  putting 

x'  =  ±  l/V2-.fl'2sin247, 

x''=±V/r"*—  #2sin24/', 
we  get 

Ax'  —  Bri'x"  =  nR  sin  (A  —  O)  —  R'  sin  (A  —  0') 

-f  n"R"  sin  (A  —  O")  —  AR'  cos  V  +  n"BR"  cos  *", 
ZtoV  —  Cx"  =  n'R'  sin  (X"  —  0')  —  jR"  sin  (A"'  —  0") 

-j-  n'"R"'  sin  (A'"  —  0'")  —  n'DR'  cos  4'  +  CR"  cos  4,". 

Let  us  now  put 

B  -h'  D       V> 

A      ht  C=h> 
or 

,  _  cos  /3"  sin  (A"  —  A)  ,  _  cos  ff  sin  (rr—  AQ 
"  cos  f  sin  (A'  —  A)  '  ~  cos  jt'  sin  (A'"—  A")' 


and  we  have 

x'  =  Kn"x"  -f  ndT  —  of  +  n'V, 
a;"  =  A"ny  +  n'"d"  —  a"  -f  nV. 

These  equations  will  serve  to  determine  xr  and  x",  and  hence  rf  and 
rr/,  as  soon  as  the  values  of  nf  nf,  nff,  and  n"1  are  known. 

96.  In  order  to  include  terms  of  the  second  order  in  the  values  of 
n  and  n",  we  have,  from  the  equations  (26)3, 


and,  putting 
these  give 


ORBIT   FROM    FOUR   OBSERVATIONS.  285 


Lot  us  now  put 

T'"  =  k  cr  —  o,         v  =  £  (*"'  —  o» 


and,  making  the  necessary  changes  in  the  notation  in  equations  (26)s, 
we  obtain 

,r-"(r0'  +  r)        ,  T"'  (T"*+  T"'T  -  1«)    dr"       \ 

~*  —  fn.  --  «  -  i^j  ---  ar*"f« 

r  **  at      i        ,.„ 

I    .r(r°'  +  ^'")    ,    .r(r'  +  rr'"-r'"0    dr" 

fs  —  ^^  ~&7r~     "df 

From  these  we  get,  including  terms  of  the  second  order, 


and  hence,  if  we  put 

P"  =  ~,  «"  =  (»'  +  «'"-!)/'«,  (17) 

we  shall  have,  since  r0'  =  r  +  r'/r, 


=   n. 

When  the  intervals  are  equal,  we  have 

P'  —  —  P"  —  — 

=  777  -pm 

and  these  expressions  may  be  used,  in  the  case  of  an  unknown  orbit, 
for  the  first  approximation  to  the  values  of  these  quantities. 
The  equations  (13)  and  (17)  give 


and,  introducing  these  values,  the  equations  (12)  become 


286  THEORETICAL   ASTEOKOMY. 

*  =  W-pr  (  1  +  %  }  (*V  +  P'd'  +  c')  - 

1  ff'\ 

x"  =  TTpr  (  1  +  ;*  )  (A'V  +  p"d"  +  c"} 

Let  us  now  put 


P'cf+c' 


An 

—  c"  -f 

C0    »  1        I        TV/     J 


(21; 


1    +    P"  1    + 

and  we  shall  have 


(22) 
+e0")-a". 

We  have,  further,  from  equations  (10), 

r''=(*/'+P'2sinV)', 
rf;«=(a/'»+JB"«sinV/)f, 

If  we  substitute  these  values  of  r'3  and  r"z  in  equations  (22),  the  two 
resulting  equations  will  contain  only  two  unknown  quantities  x'  and 
x".  when  P',  P",  §',  and  Q"  are  known,  and  hence  they  will  be 
sufficient  to  solve  the  problem.  But  if  we  effect  the  elimination  of 
either  of  the  unknown  quantities  directly,  the  resulting  equation 
becomes  of  a  high  order.  It  is  necessary,  therefore,  in  the  numerical 
application,  to  solve  the  equations  (22)  by  successive  trials,  which 
may  be  readily  effected. 

If  z'  represents  the  angle  at  the  planet  between  the  sun  and  the 
earth  at  the  time  of  the  second  observation,  and  z"  the  same  angle  at 
the  time  of  the  third  observation,  we  shall  have 

,  _  E'  sin  4/ 
~' 


(24) 
,,_' 

sin  z" 
Substituting  these  values  of  rf  and  r"  in  equations  (10),  we  get 

*'=r>™z>  (25) 

x"  =  r"  cos  z  ", 
and  hence 


ORBIT   FKOM    FOUR   OBSERVATIONS.  287 

,       R'  sin  4,' 
tan  z==  --  -,  -  , 

(26) 


by  means  of  which  we  may  find  z'  and  z"  as  ,soon  as  xf  and  x"  shall 
have  been  determined  ;  and  then  r'  and  r"  are  obtained  from  (24)  or 
(25).  The  last  equations  show  that  when  xr  is  negative,  zr  must  be 
greater  than  90°,  and  hence  that  in  this  case  r'  is  less  than  Rf. 

In  the  numerical  application  of  equations  (22),  for  a  first  approxi- 
mation to  the  values  of  xr  and  x",  since  Q'  and  Q"  are  quantities  of 
the  second  order  with  respect  to  r  or  rr//,  we  may  generally  put 


and  we  have 

*' 

*" 
or,  by  elimination, 

, 


1  -/'/" 


1  —  /'/" 


With  the  approximate  values  of  x'  and  x"  derived  from  these  equa- 
tions, we  compute  first  rr  and  r"  from  the  equations  (26)  and  (24), 
and  then  new  values  of  x'  and  x"  from  (22),  the  operation  being 
repeated  until  the  true  values  are  obtained.  To  facilitate  these  ap- 
proximations, the  equations  (22)  give 


~/'(;+f ).  >; 

*/  =  .     "  "^ 


Let  an  approximate  value  of  xf  be  designated  by  #0',  and  let  the 
value  of  x"  derived  from  this  by  means  of  the  first  of  equations  (27) 
be  designated  by  x0".  With  the  value  of  x0"  for  x"  we  derive  a 
new  value  of  xf  from  the  second  of  these  equations,  which  we  denote 
by  a?/.  Then,  recomputing  x"  and  x',  we  obtain  a  third  approximate 
value  of  the  latter  quantity,  which  may  be  designated  by  a?,';  and, 
if  we  put 

a?/  —  x0'  =  aQ,  #,'  —  xj  —  «0', 


288  THEORETICAL   ASTRONOMY. 

we  shall  have,  according  to  the  equation  (67)3,  the  necessary  changes 
being  made  in  the  notation, 

z'  =  < ?^-  =  x; PL-.  (28) 

<—  «o  «o  — «o 

The  value  of  x'  thus  obtained  will  give,  by  means  of  the  first  of 
equations  (27),  a  new  value  of  x",  and  the  substitution  of  this  in  the 
last  of  these  equations  will  show  whether  the  correct  result  has  been 
found.  If  a  repetition  of  the  calculation  be  found  necessary,  the 
three  values  of  x'  which  approximate  nearest  to  the  true  value  will, 
by  means  of  (28),  give  the  correct  result.  In  the  same  manner,  if 
we  assume  for  x"  the  value  derived  by  putting  Qf  =  0  and  Q"  =  0, 
and  compute  xf,  three  successive  approximate  results  for  x"  will 
enable  us  to  interpolate  the  correct  value. 

When  the  elements  of  the  orbit  are  already  approximately  known, 
the  first  assumed  value  of  x'  should  be  derived  from 


instead  of  by  putting  Qf  and  Q"  equal  to  zero. 

97.  It  should  be  observed  that  when  A'  =  A  or  A'"  =  A",  the  equa- 
tions (22)  are  inapplicable,  but  that  the  original  equations  (7)  give, 
in  this  case,  either  p"  or  p'  directly  in  terms  of  n  and  nff  or  of  n' 
and  n'"  and  the  data  furnished  by  observation.  If  we  divide  the 
first  of  equations  (22)  by  hf,  we  have 


The  equations  (21)  give 

f  1  /»'      P  17  ~r~  17 


h'  ~~  1  -I-  P"  A'  ~       1  +  P' 

and  from  (11)  we  get 

of  _  R'  cos  V       B  sin  (I  —  Q') 
A'  ~         A'  B 

1  =  tf'cos*"  +  ^"sin^-Q"),  (29; 

d'      J?sin(A  —  O) 


A'~  £ 

Then,  if  we  put 


ORBIT   FROM   FOUR   OBSERVATIONS. 

its  value  may  be  found  from  the  results  for  —  and  ^  derived  by 
means  of  these  equations,  and  we  shall  have 


When  A'  =  I,  we  have  h'  ==  oo,  and  this  formula  becomes 
0  =(  1  +  -J  )  (*"  +  C0')  -  £  d  +  P'), 

the  value  of  j-,  being  given  by  the  first  of  equations  (29)      This 

equation  and  the  second  of  equations  (22)  are  sufficient  to  determine 
xf  and  x"  in  the  special  case  under  consideration. 

The  second  of  equations  (22)  may  be  treated  in  precisely  the  same 
manner,  so  that  when  Xffr  =  X",  it  becomes 


o 


=( 


and  this  must  be  solved  in  connection  with  the  first  of  these  equations 
in  order  to  find  xf  and  xff. 

98.  As  soon  as  the  numerical  values  of  xr  and  xn  have  been 
derived,  those  of  rf  and  r"  may  be  found  by  means  of  the  equations 
(26)  and  (24).  Then,  according  to  (41)4,  we  have 

X*&  MO 


The  heliocentric  places  are  then  found  from  pf  and  p"  by  means  of 
the  equations  (71)3,  and  the  values  of  r'  and  r"  thus  obtained  should 
agree  with  those  already  derived.  From  these  places  we  compute 
the  position  of  the  plane  of  the  orbit,  and  thence  the  arguments  of 
the  latitude  for  the  times  tf  and  t". 

The  values  of  r',  r",  u',  u",  n,  n",  nf,  and  n"'  enable  us  to  deter- 
mine r,  r"r,  ut  and  u"f.     Thus,  we  have 


and,  from  the  equations  (1)  and  (3)3, 

19 


MJK)  THEORETICAL   ASTRONOMY. 


[rr"]    = 


Therefore, 

n 

r  sin  (uf  —  u)      =  —  r"  sin  (u"  —  w')» 
n 

r  sin  (u"  —  u)      =  -  r'  sin  (it"  —  it'), 

(32) 
i>"  sin  («"'—  «")  =  4?  r'  sin  («"  —  «'), 

71 

r'"  sin  (t*'"  —  *')  =  ^7  r"  sin  (u"  —  «')• 

From  the  first  and  second  of  these  equations,  by  addition  and  sub- 
traction, we  get 

r  sin  ((uf  —  u)  -J-  3  (w"  —  w'))  =  —  —  sm  2  V  —  w')» 

/-re»'V"  (33) 

r  cos  ((i/  —  u)  -f-  2  (w"  —  i*'))  = cos  J  (w"  —  it'), 

71 

from  which  we  may  find  r,  ur  —  it,  and  u  =  ur  —  (uf  —  it). 

In  a  similar  manner,  from  the  third  and  fourth  of  equations  (32), 
we  obtain 


rm  sin  ((um  —  t*")  +  J  (it"  —  i/))  =        V      sin  J  (i*"  -  w'), 

,/w  (34) 

r'"  cos  ((i*w  —  M")  +  A  (w"  —  u'))  =  - — m^-  cos  J  (w"  —  w'), 

71 

from  which  to  find  r'"  and  it'". 

When  the  approximate  values  of  r,  rf,  r",  rf",  and  w,  it',  it",  ufn 
have  been  found,  by  means  of  the  preceding  equations,  from  the 
assumed  values  of  P',  P",  Qf,  and  Q",  the  second  approximation  to 
the  elements  may  be  commenced.  But,  in  the  case  of  an  unknown 
orbit,  it  will  be  expedient  to  derive,  first,  approximate  values  of 
and  r",  using 

p'  —  _!_,  p"  =  _L, 

and  then  recompute  Pf  and  P"  by  means  of  the  equations  (14)  and 


ORBIT   FROM   FOUR   OBSERVATIONS.  291 

(18),  before  finding  uf  and  u".     The  terms  of  the  second  order  will 
thus  be  completely  taken  into  account  in  the  first  approximation. 

99.  If  the  times  of  observation  have  not  been  corrected  for  the 
time  of  aberration,  as  in  the  case  of  an  orbit  wholly  unknown,  this 
correction  may  be  applied  before  the  second  approximation  to  the 
elements  is  effected,  or  at  least  before  the  final  approximation  is  com- 
menced. For  this  purpose,  the  distances  of  the  body  from  the  earth 
for  the  four  observations  must  be  determined  ;  and,  since  the  curtate 
distances  pf  and  pff  are  already  given,  there  remain  only  p  and  p'tf  to 
be  found.  If  we  eliminate  pf  from  the  first  two  of  equations  (3),  the 
result  is 


nR  sin  (A'—  Q)  —  R  sin  (A'—  Q')  -f  n"  R"  sin  (X  —  0"). 


and,  by  eliminating  p"  from  the  last  two  of  these  equations,  we  also 
obtain 

,„        ,  n'  sin  (A"  -A') 

'    :  =  /V'sm(A'"-/') 

n1  R'  sin  (A"  -  Q')  —  R"  sin  (A"  —  Q  ")  -f  ri"  R"f  sin  (A"  —  0  "') 
n'"  sin  (A'"—  /I") 

by  means  of  which  p  and  p"f  may  be  found.     The  combination  of 
the  first  and  second  of  equations  (3)  gives 


COS  (X  — 

nR  cos  a  —  Q)  —  R'  cos  (A  —  0')  +  n" R"  cos  (A  —  0") 


/o  =  ^  cos  (A'  —  A)  —  2_£-  cos  (A"  —  A)  (37) 

n  n 


+ 
and  from  the  third  and  fourth  we  get 

/>''cos(r'_A'o-$' 

n'  R1  cos  (A'"—  0')  —  R'  cos  (A'"—  0")  -f  n'"  R'"  cos  (A'"—  Q '") 


.'"  =  ^7  cos  (A'"  -  A")  —  ^  cos  (A'"  —  A')  (38) 


Further,  instead  of  these,  any  of  the  various  formulae  which  have 
been  given  for  finding  the  ratio  of  two  curtate  distances,  may  be 
employed;  but,  if  the  latitudes  /?,  /9',  &c.  are  very  small,  the  values 
of  p  and  p"'  which  depend  on  the  differences  of  the  observed  longi- 
tudes of  the  body  must  be  preferred. 


292  THEORETICAL    ASTRONOMY. 

T1ie  values  of  pf  and  prrf  may  also  be  derived  by  computing  the 
heliocentric  places  of  the  body  for  the  times  t  and  t'n  by  means  of 
the  equations  (82)1?  and  then  finding  the  geocentric  places,  or  those 
which  belong  to  the  points  to  which  the  observations  have  been 
reduced,  by  means  of  (90)b  writing  p  in  place  of  Jcos/9.  This 
process  affords  a  verification  of  the  numerical  calculation,  namely, 
the  values  of  X  and  X'fr  thus  found  should  agree  with  those  furnished 
by  observation,  and  the  agreement  of  the  computed  latitudes  ft  and 
ft'"  with  those  observed,  in  case  the  latter  are  given,  will  show  how 
nearly  the  position  of  the  plane  of  the  orbit  as  derived  from  the 
second  and  third  observations  represents  the  extreme  latitudes.  If 
it  were  not  desirable  to  compute  X  and  X"  in  order  to  check  the 
calculation,  even  when  ft  and  ftflf  are  given  by  observation,  we  mighi 
derive  p  and  p"r  from  the  equations 

p    =  r  sin  u  sin  i  cot  y5, 
p'"  =  r'"  sin  u"f  sin  i  cot  p", 

when  the  latitudes  are  not  very  small. 

In  the  final  approximation  to  the  elements,  and  especially  when 
the  position  of  the  plane  of  the  orbit  cannot  be  obtained  with  the 
required  precision  from  the  second  and  third  observations,  it  will  be 
advantageous,  provided  that  the  data  furnish  the  extreme  latitudes 
ft  and  ft"f,  to  compute  p  and  pf"  as  soon  as  pf  and  pff  have  been 
found,  and  then  find  I,  lf",  6,  and  b'"  directly  from  these  by  means 
of  the  formulae  (71)3.  The  values  of  &  and  i  may  thus  be  obtained 
from  the  extreme  places,  or,  the  heliocentric  places  for  the  times  tr 
and  t'"  being  also  computed  directly  from  p'  and  p",  from  those 
which  are  best  suited  to  this  purpose.  But,  since  the  data  will  be 
more  than  sufficient  for  the  solution  of  the  problem,  when  the  extreme 
latitudes  are  used,  if  we  compute  the  heliocentric  latitudes  bf  and  b'" 
from  the  equations 

tan  b'  =  tan  i  sin  (I'  —  &  ), 
tan  b"  =  tan  i  sin  (I"  —  &), 

they  will  not  agree  exactly  with  the  results  obtained  directly  from  p1 
and  /?",  unless  the  four  observations  are  completely  satisfied  by  the 
elements  obtained.  The  values  of  r'  and  rn ',  however,  computed 
directly  from  pr  and  p"  by  means  of  (71)3,  must  agree  with  those 
derived  from  xr  and  xn. 

The  corrections  to  be  applied  to  the  times  of  observation  on  account 


ORBIT   FROM    FOUR   OBSERVATIONS.  293 

of  aberration  may  now  be  found.     Thus,  if  tw  t0f,  tQ",  and  tQ'"  are 
the  uncorrected  times  of  observation,  the  corrected  values  will  be 

t    =  t0  —  Cp  sec/5, 


wherein  log  C==  7.760523,  and  from  these  we  derive  the  corrected 
values  of  r,  r',  r",  r'",  and  r0'. 

100.  To  find  the  values  of  P',  P",  Q',  and  Q",  which  will  be 
exact  when  r,  r',  r",  rf//,  and  w,  u'9  u",  u'"  are  accurately  known,  we 
have,  according  to  the  equations  (47)4  and  (51)4,  since  Qf  =  |§, 


_  _    _ 

~"  *  ss"  '  rr"  cos  J  (t*"  —  w')  cos  i  (t*"  —  M)  cos  j  («'  —  «)' 

In  a  similar  manner,  if  we  designate  by  sfff  the  ratio  of  the  sector 
formed  by  the  radii-vectores  rrr  and  r'"  to  the  triangle  formed  by 
the  same  radii-vectores  and  the  chord  joining  their  extremities,  we 
find 


_  i  rr'" r^ 

V     :  ~  2  ^777  '  ry/,  CoS  l  (>'"  —  M")  COS  i  (U'"  —  t*')  COS  J  (u"  —  1*')' 

The  formulae  for  finding  the  value  of  s'rr  are  obtained  from  those  for 
s  by  writing  //r/,  fff,  G"f,  &c.  in  place  of  /,  /-,  (r,  &c.,  and  using 
r",  r'",  w//;  —  u"  instead  of  r',  r",  and  it"  —  M',  respectively. 

By  means  of  the  results  obtained  from  the  first  approximation  to 
the  values  of  P',  P",  Q',  and  §",  we  may,  from  equations  (41)  and 
(42),  derive  new  and  more  nearly  accurate  values  of  these  quantities, 
and,  by  repeating  the  calculation,  the  approximations  to  the  exact 
values  may  be  carried  to  any  extent  which  may  be  desirable.  When 
three  approximate  values  of  P'  and  §',  and  of  P"  and  §",  have 
been  derived,  the  next  approximation  will  be  facilitated  by  the  use 
of  the  formulae  (82)4,  as  already  explained. 

When  the  values  of  P',  P",  Qf,  and  Q"  have  been  derived  with 
sufficient  accuracy,  we  proceed  from  these  to  find  the  elements  of  the 
orbit.  After  &,  i,  r,  rf,  r" ,  r'",  u,  u',  u" ',  and  uf/f  have  been  found, 
the  remaining  elements  may  be  derived  from  any  two  radii-vectores 


294  THEORETICAL   ASTRONOMY. 

and  the  corresponding  arguments  of  the  latitude.  It  will  be  most 
accurate,  however,  to  derive  the  elements  from  r,  r/r/,  u,  and  u"f. 
If  the  values  of  P',  P",  Q',  and  Q"  have  been  obtained  with  great 
accuracy,  the  results  derived  from  any  two  places  will  agree  with 
those  obtained  from  the  extreme  places. 
In  the  first  place,  from 


cos  G0  =  sin  £  (u"f  —  u),  (43) 

sin  YQ  sin  O0  =  cos  £  (V"  —  w)  cos  2/0, 
cos  r0  =  cos  |  (u'"  —  u)  sin  2/0, 


we  find  p0  and  6r0.     Then  we  have 


r'")S 


"to  (44) 


from  which,  by  means  of  Tables  XIII.  and  XIV.,  to  find  s0  and  a?0. 
We  have,  further, 


and  the  agreement  of  the  value  of  p  thus  found  with  the  separate 
results  for  the  same  quantity  obtained  from  the  combination  of  any 
two  of  the  four  places,  will  show  the  extent  to  which  the  approxima- 
tion to  P;,  P")  Qf,  and  Q"  has  been  carried.  The  elements  are  now 
to  be  computed  from  the  extreme  places  precisely  as  explained  in  the 
preceding  chapter,  using  r'"  in  the  place  of  r"  in  the  formula  there 
given  and  introducing  the  necessary  modifications  in  the  notation, 
which  have  been  already  suggested  and  which  will  be  indicated  at 
once. 

101.  EXAMPLE.  —  For  the  purpose  of  illustrating  the  application 
of  the  formulae  for  the  calculation  of  an  orbit  from  four  observations, 
let  us  take  the  following  normal  places  of  Eurynome  ®  derived  by 
comparing  a  series  of  observations  with  an  ephemeris  computed  from 
approximate  elements. 

Greenwich  M.  T.  a  {, 

1863  Sept.  20.0  14°  30'  35".6  +    9°  23'  49".7, 
Dec.     9.0  9    54  17  .0                  2    53  41  .8, 

1864  Feb.      2.0  28    41  34  .1                   962  .8, 
April  30.0  74    29  58  .9  -f-  19    35  41  .5. 


NUMERICAL   EXAMPLE.  295 

These  normals  give  the  geocentric  places  of  the  planet  referred  to  the 
mean  equinox  and  equator  of  1864.0,  and  free  from  aberration.  For 
the  mean  obliquity  of  the  ecliptic  of  1864.0,  the  American  Nautical 
Almanac  gives 

e  =  23°  27'  24".49, 

and,  by  means  of  this,  converting  the  observed  right  ascensions  and 
declinations,  as  given  by  the  normal  places,  into  longitudes  and  lati- 
tudes, we  get 


Greenwich  M.  T. 
1863  Sept.  20.0 
Dec.      9.0 
1864  Feb.      2.0 
April  30.0 

A 
16°  59'    9".42 
10    14  17  .57 
29    53  21  .99 
75    23  46  .90 

ft 
4-  2°  56'  44".58, 
—  1    15  48  .82, 
2    29  57  .38, 
—  3     4  44  .49. 

These  places  are  referred  to  the  ecliptic  and  mean  equinox  of  1864.0, 
and,  for  the  same  dates,  the  geocentric  latitudes  of  the  sun  referred 
also  to  the  ecliptic  of  1864.0  are 

+  0".60,  -f-0".53,  -f-0".36,  +  0".19. 

For  the  reduction  of  the  geocentric  latitudes  of  the  planet  to  the 
point  in  which  a  perpendicular  let  fall  from  the  centre  of  the  earth 
to  the  plane  of  the  ecliptic  cuts  that  plane,  the  equation  (6)4  gives  the 
corrections  —  0".57,  —  0".38,  —  0".18,  and  —  0".07  to  be  applied  tc 
these  latitudes  respectively,  the  logarithms  of  the  approximate  dis- 
tances of  the  planet  from  the  earth  being 

0.02618,  0.13355,  0.29033,  0.44990. 

Thus  we  obtain 

t    =     0.0,  A    =  16°  59'    9".42,  ft    =  +  2°  56'  44".01, 

if  =   80.0,  X  =10    14  17  .57,  f  =  —  1    15  49  .20, 

«"  =  135.0,  A"  =  29    53  21  .99,  f  =  —  2    29  57  .56, 

f"  =  223.0,  r  =  75    23  46  .90,  jfn  =  —  3     4  44  .56  ; 

and,  for  the  same  times,  the  true  places  of  the  sun  referred  to  the 
mean  equinox  of  1864.0  are 

0    =177°    0'58".6,  logE    =0.0015899. 

0'  =256    58  35.9,  log#   =9.9932638, 

0"  =312    57  49  .8,  log-R"  =9.9937748, 

0'"=   40    21  26.8,  log  #"  =  0.0035149, 


296 


THEORETICAL   ASTRONOMY. 


From  the  equations 

tan/5' 


tan  wf  =    . 


tan  to"  = 


.    ,.,        -j^> 
sm(A'—  O) 

tan£" 

.    ,  „  '          , 
sm(A"—  O  ) 


tan  (A'—  Q') 

tan  *   =  --  -  -  7=^1 

cosw/ 

„      tan(A"—  O") 
tan  V  =  -  -  -  „      \ 
cosw" 


we  obtain 

4/  =  113°  15'  20".10, 
4,"=   76    5617.75, 


cos  V)  =  9.5896777n, 
log  (R  sin  4,')  =9.9564624, 
log  (#'  cos  4")  =  9.3478848, 
log  (R"  sin  *")  =  9.9823904. 


The  quadrant  in  which  ty  must  be  taken,  is  indicated  by  the  condi- 
tion that  cosij/  and  cos  (A'  —  O')  must  have  the  same  sign.  The 
same  condition  exists  in  the  case  of  i//'»  Then,  the  formulaB 


A  =  cos  ff  sin  (^  —  A), 
C  =  cos  /9"  sin  (A'"—  A"), 

^-A' 

A-  h, 


B  =  cos  p'  sin  (A"  —  jl), 
Z>  =  cos  f  sin  (A'"—  A'), 


,, 


—  O) 


0 


,ft 


"—  Q'") 


give  the  following  results  :  — 

log  A  =  9.0699254n, 
log  B  =  9.3484939, 
log  h'  =  0.2785685n, 
log  a!  =  0.8834880n, 
log  d  =  0.9012910n, 
log  d'  =  0.4650841, 


log  C  =  9.8528803, 
log  D  =  9.9577271, 
log  h"  =  0.1048468, 
log  a"  =  9.9752915n, 
log  c"  =  9.7267348w, 
log  d"  =  9.9096469n. 


We  are  now  prepared  to  make  the  first  hypothesis  in  regard  to  the 
values  of  P',  §',  Pf/,  and  Q".  If  the  elements  were  entirely  un- 
known, it  would  be  necessary,  in  the  first  instance,  to  assume  for  these 
quantities  the  values  given  by  the  expressions 


NUMERICAL   EXAMPLE.  297 


then  approximate  values  of  r'  and  r"  are  readily  obtained  by  means 
of  the  equations  (27),  (26),  and  (24)  or  (25).  The  first  assumed 
value  of  x'  to  be  used  in  the  second  member  of  the  first  of  equations 
(27),  is  obtained  from  the  expression  which  results  from  (22)  by 
putting  Qf  =  0  and  Q"  =  0,  namely, 

x'  =  * 


after  which  the  values  of  xf  and  x"  will  be  obtained  by  trial  from 
(27).  It  should  be  remarked,  further,  that  in  the  first  determination 
of  an  orbit  entirely  unknown,  the  intervals  of  time  between  the  ob- 
servations will  generally  be  small,  and  hence  the  value  of  xf  derived 
from  the  assumption  of  Qf  =  0  and  Q"  —  0  will  be  sufficiently  ap- 
proximate to  facilitate  the  solution  of  equations  (27). 

As  soon  as  the  approximate  values  of  rf  and  r"  have  thus  been 
found,  those  of  P'  and  P"  must  be  recomputed  from  the  expressions 


With  the  results  thus  derived  for  P'  and  P",  and  with  the  values  of 
Qf  and  Q"  already  obtained,  the  first  approximation  to  the  elements 
must  be  completed. 

When  the  elements  are  already  approximately  known,  the  first 
assumed  values  of  P',  P",  Q',  and  Qf/  should  be  computed  by  means 
of  these  elements.  Thus,  from 

_rV'smQ/'  —  t/)  „         r/sinQ/  —  1>) 

~  rr"  sin  («"  —  v)  '  ~  rr"  sin  (v"  —  v)  ' 

,rV"sin(y"—  f/Q  w_  rV;sin(^  —  i;Q 

~ 


r  r    sin  (v    —  v)  TT    sin  (v    —  v ) 

we  find  n,  n'3  nrf,  and  n"'.     The  approximate  elements  of  Eurynom* 

givo 

v    =  322°  55'    9".3,  logr    =0.308327, 

it  =353    19  26  .3,  log/  =0.294225, 

v"=   14    45     8.5,  log/' =0.296088, 

v'"=   47    2332.8,  log  /"  =  0.317278, 


298  THEORETICAL   ASTRONOMY. 

and  hence  we  obtain 

log  n  =  9.653052,  log  n"  =  9.806836, 

log  n1  =  9.825408,  logn'"  =  9.633171. 

Then,  from 


P"  =      , 

we  get 

log  P'  =  9.846216,  log  Qf  =  9.840771, 

log  P"=  9.807763,  log  Qf'  =  9.882480. 

The  values  of  these  quantities  may  also  be  computed  by  means  of  the 
equations  (41)  and  (42). 
Next,  from 

,  _  P'd'  +  c'  ,  _        hf 

C°  ~~  =  1  +  P"  *     ~  1  -f-  P'' 

,_P»d"+c"  h" 

co  =  =  x  +  Pn  >  J    -  fqrp> 

we  find 

log  c0'  =  0.541344n,  log/  =  0.047658., 

log  CQ"  =  9.807665n,  log/"  =  9.889385. 

Then  we  have 

' 


*=  *"+""„    < 


,  ,, 

tan  /  =  —  -—-,  tan  z"  = 

,_R'  sin  V  _      a/  ,,_jrsinV_      a" 

sin/          cos/'  sin/7          cos/" 

from  which  to  find  r'  and  rrr.     In  the  first  place,  from 


x'  =  v1  —  jR"sinV, 
we  obtain  the  approximate  value 

log  xf  =  0.242737. 

Then  the  first  of  the  preceding  equations  gives 

log  *"  =  0.237687. 


NUMERICAL   EXAMPLE.  299 

From  this  we  get 

z"  =  29°  3'  11".  7,  log  r"  =  0.296092  ; 

and  then  the  equation  for  x'  gives 

logo;'  =  0.242768. 
Hence  we  have 

z'  =  27°  20'  59".6,  log  r'  =  0.294249  ; 

and,  repeating  the  operation,  using  these  results  for  xr  and  r',  we  get 
log  x"  =  0.237678,  log  of  =  0.242757. 

The  correct  value  of  log  a?'  may  now  be  found  by  means  of  equation 
(28).     Thus,  in  units  of  the  sixth  decimal  place,  we  have 

«0  =  242768  —  242737  =  +  31,  aQ'  =  242757  —  242768  =  —  11, 

and  for  the  correction  to  be  applied  to  the  last  value  of  log  x',  in 
units  of  the  sixth  decimal  place, 


Therefore,  the  corrected  value  is 

log  af=  0.242760, 
and  from  this  we  derive 

log  a"  =  0.237681. 

These  results  satisfy  the  equations  for  x'  and  x",  and  give 

z'  =  27°  21'    1".2,  log/  =  0.294242, 

z"  =  29      3  12  .9,  log  r"  =  0.296087. 

To  find  the  curtate  distances  for  the  first  and  second  observations, 
the  formulae  are 


oo  o,,  =  - 

sin  z'  sm  z" 

which  give 

log  p'  =  0.133474,  log  p"  =  0.289918. 

Then,  by  means  of  the  equations 


300  THEORETICAL   ASTRONOMY. 

r'  cos  V  cos  (f  —  00     =  p'  cos  (X  —  ©0  —  K9 
r'  cos  V  sin  (?  —  ©')     =  p'  sin  (A'  —  QO, 
/  sin  V  =  p'  tan  p, 

r"  cos  ft"  cos  (r  —  Q'O  =  P"  cos  (A"  —  0")  —  J2", 
r"  cos  b"  sin  (/"  —  Q")  =  p"  sin  (A"  —  ©"), 
r"  sin  ft"  =  //'tan/S", 

we  find  the  following  heliocentric  places  : 

r  =  37°  35'  26".4,  log  tan  ft'  =  8.182861n,  log  /  =  0.294243, 

r  =  58    5815.3,  logtanft"  =  8.634209n,  log  r"  =  0.296087. 

The  agreement  of  these  values  of  log  r'  and  log  r"  with  those  obtained 
directly  from  x'  and  x"  is  a  partial  proof  of  the  numerical  calcula- 
tion. 

From  the  equations 

tan  i  sin  (£  (I"  -f  /')  —  £  )  =  J  (tan  ft"  +  tan  ftO  sec  j  (r  —  7), 
tan  i  cos  Q  (J1  -f  I')  —  &  )  =  J  (tan  ft"  —  tan  ftO  cosec  i  (/"  —  0, 

tan  (^—  &0  „      tan(r—  ft) 

tan  w  =  -  -  -  r-^—  ,  tan  u  = 


cos  i  cos  t 

we  obtain 

&  =  206°  42'  24".0,  i    =     4°  36'  47".2, 

ur  =190    55     6  .6  u"  =  212    20  53  .5. 

Then,  from 


we  get 

log  n"  =  9.806832,  logw    =9.653048, 

log  n'  =  9.825408,  log  n'"  =  9.633171, 

and  the  equations 

r  sin  ((ur  -  u)  +  i  (u"  -  w'))  =  /+J'V'  sin  J  (i*"  -  «0, 

r  cos  ((i*'  -  u)  +  i  (w"  -  tO)  =  ^=          cos  i  (u"  -  tif  ), 


r"'  sin  ((*"'  -  w'O  +  J  (u"  -  wO)  -          ,       sin  j  (t»"  -  wO, 


r"'  cos  ((u'"  -  i/O  -f  i  (u"  -  1*0)  =  cos  i  (u"  - 


NUMERICAL   EXAMPLE.  301 

give 

logr    =  0.308379,  u    =  160°  30;  57".6, 

log  r"'  =  0.317273,  t*'"=244    5932.5. 

Next,  by  means  of  the  formulae 

tan  (I  —  &  )    =  cos  i  tan  u,  tan  6    =  tan  i  sin  (/  —  ft  ), 

tan  (Y"  —  £  )  =  cos  i  tan  w'",  tan  6'"  =  tan  i  sin  (£'"  —  ft  ), 

^  cos  (A  —  o  )          =  f  cos  6  cos  (7  —  0)  +  -K, 
jo  sin  (A  —  Q  )          =  r  cos  6  sin  (7  —  O), 
/o  tan  /?  =  r  sin  &  ; 

p'»  Cos  (A"'  —  O'")  =  r'"  cos  6'"  cos  (f"—  ©"0  +  12"', 
p'»  sin  (A'"  —  O'")  =  r'"  cos  6"'  sin  (f"  —  0'"), 
//"tan/5"'  =  r"'sin&'", 

we  obtain 

J  =        7°  16'  51".8,  r  =      91°  37'  40".0, 

b  =  +    1    32  14  .4,  6"'  =  —    4    10  47  .4, 

A  =      16    59     9  .0,  /"  —      75    23  46  .9, 

/?  =  +    2    56  40  .1,  /?'"  =  —    3      4  43  .4, 

log  P  =  0.025707,  log  p'"  =  0.449258. 

The  value  of  X"f  thus  obtained  agrees  exactly  with  that  given  by 
observation,  but  /  differs  Or/.4  from  the  observed  value.  This  differ- 
ence does  not  exceed  what  may  be  attributed  to  the  unavoidable 
errors  of  calculation  with  logarithms  of  six  decimal  places.  The 
differences  between  the  computed  and  the  observed  values  of  /9  and 
ft"  show  that  the  position  of  the  plane  of  the  orbit,  as  determined 
by  means  of  the  second  and  third  places,  will  not  completely  satisfy 
the  extreme  places. 

The  four  curtate  distances  which  are  thus  obtained  enable  us,  in 
the  case  of  an  orbit  entirely  unknown,  to  complete  the  correction  for 
aberration  according  to  the  equations  (40). 

The  calculation  of  the  quantities  which  are  independent  of  P', 
Pffj  Qfj  and  Q/f,  and  which  are  therefore  the  same  in  the  successive 
hypotheses,  should  be  performed  as  accurately  as  possible.  The 

s* 

value  of  -^->  required   in   finding   x"  from   x',  may  be   computed 
directly  from 


•jf  . 

the  values  of  77  and  77  being  found  by  means  of  the  equations  (29); 


302  THEORETICAL   ASTRONOMY. 

and  a  similar  method  may  be  adopted  in  the  case  of  j,r>     Further, 

in  the  computation  of  xf  and  x",  it  may  in  some  cases  be  advisable 
to  employ  one  or  both  of  the  equations  (22)  for  the  final  trial.  Thus, 
in  the  present  case,  x"  is  found  from  the  first  of  equations  (27)  by 
means  of  the  difference  of  two  larger  numbers,  and  an  error  in  the 
last  decimal  place  of  the  logarithm  of  either  of  these  numbers  affects 

in  a  greater  degree  the  result  obtained.     But  as  soon  as  r"  is  known 

Q" 
so  nearly  that  the  logarithm  of  the  factor  1  -f  -^  remains  unchanged, 

the  second  of  equations  (22)  gives  the  value  of  x"  by  means  of  the 
sum  of  two  smaller  numbers.  In  general,  when  two  or  more  for- 
mulae for  finding  the  same  quantity  are  given,  of  those  which  are 
otherwise  equally  accurate  and  convenient  for  logarithmic  calculation, 
that  in  which  the  number  sought  is  obtained  from  the  sum  of  smaller 
numbers  should  be  preferred  instead  of  that  in  which  it  is  obtained 
by  taking  the  difference  of  larger  numbers. 

The  values  of  r,  rf,  r/f,  r'",  and  u,  ufy  uff,  ufff,  which  result  from 
the  first  hypothesis,  suffice  to  correct  the  assumed  values  of  Pf,  P", 
§',  and  Q".  Thus,  from 

"r77"  „          '/  ,„         /r777" 

sin  Y  cos  G  =  sin  ±  (u" —  u'),  sin  /'  cos  G"  =  sin  A  (ur —  u), 

sin  f  sin  G  =  cos  J  (u" —  u')  cos  2/,     sin  /'  sin  G"  —  cos  £  (ur —  u)  cos  2/', 

cos  f  =  cos  |  (u" —  u')  sin  2/,     cos  f"  =  cos  \  (V —  u)  sin  2,%", 

sin  f  cos  G'"  =  sin  I  (u"r  —  w"), 

sin  f  sin  G"'  =  cos  £  (u"f  —  u"}  cos  2/" 

cos  /"  =  cos  J  (u"'  —  u")  sin  2/" ; 

T2COS6/  _„_    T"2COS6/' 

m  = 


COS?' 

m 


r'/scos3/' 


t+j  +  ?  *+/'+* 


in  connection  with  Tables  XIII.  and  XIV.  we  find  s,  s",  and  «'". 
The  results  are 


NUMEKICAL   EXAMPLE. 


303 


log  T  =  9.9759441, 
/  =  45°    3'39".l, 
r=-10    42  55  .9, 

log  m  =  8.186217, 
log;  =  7.948097, 
log  s  =  0.0085248, 


log  r"=  0.1386714, 
7"=44°32'    1".4, 
/'—IS    13  45  .0, 
log  m"=  8.516727, 
log/'=  8.260013, 
log  «"==  0.0174621, 


log  T"'=  0.1800641, 
/"==  45°  41'  55".2, 
y"'=16   22  48  .5. 
log  m'"=  8.590596, 
log/"=  8.325365, 
log  s'"=  0.020406?. 


Then,  by  means  of  the  formulae 


a  -* 

V  -     2     ,, 


y,  = ,  nl' _£ 

2  ss"f     ry //  cog  ^  (yftr  —  un^  cog  ^  (jjin  —  ^  CQg  ^  ^/  —  %/y 

we  obtain 


log  P'  =  9.8462100, 
log  P"  =  9.8077615, 


log  §'  =  9.8407536, 
log  £"  =  9.8824728, 


with  which  the  next  approximation  may  be  completed. 

We  now  recompute  c0',  c/',  /',  /",  xf,  xn  ',  &c.  precisely  as  already- 
illustrated;  and  the  results  are 


log  c0'  =  0.5413485n, 
log/  =  0.0476614n, 
log  x'  =  0.2427528, 

zf  =  27°  21'  2".71, 
log/  =0.2942369, 
log  />'  =  0.1334635, 
log  n  =  9.6530445, 
log  n'  =  9.8254092, 


log  c0"  =  9.8076649n, 
log/"  =  9.8893851, 
log  x"  =  0.2376752, 

z"  =  29°  3'  14".09, 
logr"  =0.2960826, 
log  p"  =  0.2899124, 
log  n"  =  9.8068345, 
log  ri"  =  9.6331707. 


Then  we  obtain 

/'  =  37°  35'  27".88, 
I"  —  58    58  16  .48, 


log  tan  V  =  8.1828572n, 
log  tan  b"  =  8.6342073n, 


log/  =  0.2942369, 
log  /'  =  0.2960827. 


These  results  for  log  rf  and  log  r"  agree  with  those  obtained  directly 
from  zf  and  z",  thus  checking  the  calculation  of  ty  and  tyf  and  of 
the  heliocentric  places. 
Next,  we  derive 


206°  42'  25".89, 
55     6.27, 


t  =     4°  36'  47".20, 
2052.96, 


304  THEORETICAL   ASTRONOMY. 

and  from  uff  —  u',  r',  r",  n,  n",  nf,  and  n"f,  we  obtain 

logr    =0.3083734,  u    =  160°  30'  55".45, 

log/"  =0.3172674,  w'"=244    5931.98. 

For  the  purpose  of  proving  the  accuracy  of  the  numerical  results, 
we  compute  also,  as  in  the  first  approximation, 

/=        7°16'51".54,  /'"  =      91°  37'  41".20, 

b  =  +    1    32  14  .07,  b'"=—    4    10  47  .36, 

A=       16    59     9.38,  A'"  =      75    2346.99, 

0  =  +    2    56  39  .54,  p"=—    3     4  43  .33, 

log  P  =  0.0256960,  log  p"r  =  0.4492539. 

The  values  of  ^  and  \tn  thus  found  differ,  respectively,  only  0".04 
and  Or/.09  from  those  given  by  the  normal  places,  and  hence  the 
accuracy  of  the  entire  calculation,  both  of  the  quantities  which  are 
independent  of  P',  P"  ',  §',  and  §",  and  of  those  which  depend  on 
the  successive  hypotheses,  is  completely  proved.  This  condition, 
however,  must  always  be  satisfied  whatever  may  be  the  assumed 
values  of  P',  P",  Qf,  and  §". 
From  TJ  r'j  u,  it',  &c.,  we  derive 

log  a  =  0.0085254,        log  s"  =  0.0174637,        log  sm  =  0.0204076, 
and  hence  the  corrected  values  of  P',  P",  §',  and  Q"  become 

logP'  =  9.8462110,  log  q  =  9.8407524, 

log  P"  =  9.8077622,  log  §"  =  9.8824726. 

These  values  differ  so  little  from  those  for  the  second  approximation, 
the  intervals  of  time  between  the  observations  being  very  large,  that 
a  further  repetition  of  the  calculation  is  unnecessary,  since  the  results 
which  would  thus  be  obtained  can  differ  but  slightly  from  those 
which  have  been  derived.  We  shall,  therefore,  complete  the  deter- 
mination of  the  elements  of  the  orbit,  using  the  extreme  places. 
Thus,  from 


sin  YQ  cos  G0  =  sin  J  (utn  —  u), 

sin  YO  sin  G0  =  cos  J  (u"f  —  u)  cos  2/0, 

cos  YO  —  cos  £  (u"'  —  u)  sin  2/0, 


5  i  i 
«  ~r  Jo 


?-  _ 
" 


m0  _  ma 

°  ~~    7  ~ 


NUMERICAL  EXAMPLE.                                           305 

we  get 

log  r0  =  0.5838863,  log  tan  GQ  =  8.0521953W> 

n  =  42°  14'  30".17,  log  m0  =  9.7179026, 

log  s02  =  0.2917731,  log  x0  =  8.9608397. 

The  formula 

s0rrmsm(um—  t*)\» 


gves 

log  p  =  0.371  2401; 

and  if  we  compute  the  same  quantity  by  means  of 

grV'sm(X'—  •  iQ  \2     /  g'Vr'sinK—  u)  V     I  s'VV'sin  (u'"- 


u"}  \» 

I* 


the  separate  results  are,  respectively,  0.3712397,  0.3712418,  and 
0.3712414.  The  differences  between  these  results  are  very  small,  and 
arise  both  from  the  unavoidable  errors  of  calculation  and  from  the 
deviation  of  the  adopted  values  of  P',  Px/,  Q',  and  Q"  from  the 
limit  of  accuracy  attainable  with  logarithms  of  seven  decimal  places. 
A  variation  of  only  Or/.2  in  the  values  of  u'  —  u  and  ufff  —  uff  wil? 
produce  an  entire  accordance  of  the  particular  results. 
From  the  equations 


smi(V"  —  u)      /—T,, 
°  C°8  p  =  sin  K£"'  -£" 

P 

cos  <p  = 


we  obtain 

t  (£"'  —  E)  =  17°  35'  42".12,  log  (a  cos  <?)  =  0.3796883, 

log  cos  <?  =  9.9915518. 
The  formulae 

e  Sin  («,  -  J  (u"r  +  u))  =  -  £-—  tan  G0, 
cos  y0  Vrr'" 

e  cos  (.  -  J  («T"  +  u))  -  -  £7==  -  sec  J  (u'"  -  u), 

cos  ^0  rrr* 
give 

01  =  197°  38'  8".48,  log  e  =  log  sin  <p  =  9.2907881, 

<p  =  11°  15'  52".22,  TT  =  01  -|-  ft  =  44°  20'  34".37. 

This  result  for  ^  gives  log  cos  <p  —  9.9915521,  which  differs  only  3 
in  the  last  decimal  place  from  the  value  found  from  p  and  a  cos  <p. 
Then,  from 

20 


306  THEORETICAL   ASTRONOMY. 

p  k 

-c^sV  =JP 

the  value  of  k  being  expressed  in  seconds  of  arc,  or  log&  =  3.5500066, 
we  get 

log  a  =  0.3881359,  log  ft  =  2.9678027. 

For  the  eccentric  anomalies  we  have 

tan^jE  =  tan4O  —  o>)  tan  (45°  —  £?), 
t&n%E'  =  tani(X  —  oO  tan  (45°  —  £?>), 
tan  IE"  =  tan  J  O"  -  o»)  tan  (45°  —  '  p), 
tan  §E'"  =  tan  £  (u"r—  o»)  tan  (45°  —  £?>)• 

from  which  the  results  are 

E  =  329°  11'  46".01,  .E"  —  12°    5'  33".63, 

£'  =  354    29  11  .84,  £'"  =  39    34  34  .65. 

The  value  of  J  (E"f  —  E)  thus  derived  differs  only  0".03  from  that 
obtained  directly  from  x0. 

For  the  mean  anomalies,  we  have 


which  give 


,  M"  =  E"  —  esmE", 

=  Er  —  e  sin  E',  M'"  =  E'"  —  e  sin  E'", 


M  =  334°  55'  39".32,  M"  =   9°  44'  52".82, 

M'  =  355    33  42  .97,  JJf'"  =  32    26  44  .74. 

Finally,  if  MQ  denotes  the  mean  anomaly  for  the  epoch  T=  1864 
Jan.  1.0  mean  time  at  Greenwich,  from 

MQ  =  M  —  fi  (t  —  T)     =Mt  —  p(f—T) 
=  M"  —  fJL  (f  —  T}  =  M'"  —  p.  (f"  —  T), 

we  obtain  the  four  values 

Jf0  =  l°29'39".40 
39  .49 
39  .40 
39  .40, 

the  agreement  of  which  completely  proves  the  entire  calculation  of 
the  elements  from  the  data.  Collecting  together  the  several  results, 
we  have  the  following  elements : 


NUMERICAL   EXAMPLE.  307 

Epoch  =  1864  Jan.  1.0  Greenwich  mean  time. 
M  =     1°  29'  39".42 


= 

?=    11    15  52  .22 

log  a  =  0.3881359 
log  ;«  =  2.9678027 
A*  =  928".54447. 

102.  The  elements  thus  derived  completely  represent  the  four  ob- 
served longitudes  and  the  latitudes  for  the  second  and  third  places, 
which  are  the  actual  data  of  the  problem  ;  but  for  the  extreme  lati- 
tudes the  residuals  are,  computation  minus  observation, 

A£  =  —  4".47,  A,?'"  =  +  1".23. 

These  remaining  errors  arise  chiefly  from  the  circumstance  that  the 
position  of  the  plane  of  the  orbit  cannot  be  determined  from  the 
second  and  third  places  with  the  same  degree  of  precision  as  from 
the  extreme  places.  It  would  be  advisable,  therefore,  in  the  final 
approximation,  as  soon  as  pf,  p",  n,  n'f,  nf,  and  nrrr  are  obtained,  to 
compute  from  these  and  the  data  furnished  directly  by  observation 
the  curtate  distances  for  the  extreme  places.  The  corresponding 
heliocentric  places  may  then  be  found,  and  hence  the  position  of  the 
plane  of  the  orbit  as  determined  by  the  first  and  fourth  observations. 
Thus,  by  means  of  the  equations  (37)  and  (38),  we  obtain 

log  P  =  0.0256953,  log  p'"  =  0.4492542. 

With  these  values  of  p  and  //",  the  following  heliocentric  places  are 
obtained  : 

I  =  7°16'51".54,  log  tan  b  =8.4289064,  logr  =0.3083732, 
lf"  =  91  37  40  .96,  log  tan  b'"  =  8.8638549n,  log  r"'  =  0.3172678. 

Then  from 

tan  i  sin  (J  (f"  +  1)  —  ft)  =  J  (tan  V"  -f  tan  6)  sec  J  (f"  —  Z), 
tan  i  cos  Q  (f"  -f  0  —  &  )  =  i  (tan  b"'  —  tan  6)  cosec  J  (f»  —  I), 

we  get 

^  =  206°  42'  45".23,  i  =  4°  36'  49".76. 

For  the  arguments  of  the  latitude  the  results  are 

u  =  160°  30'   35".99,  u'"  =  244°  59'  12".53. 


398  THEORETICAL   ASTRONOMY. 

The  equations 

tan  V  =  tan  i  sin  (I1  —  Q  ), 
tan  V  =  tan  i  sin  (I"  —  £  ), 
give 

log  tan  &'  =  8.1827129^  log  tan  b"  =  8,6342104n, 

and  the  comparison  of  these  results  with  those  derived  directly  from 
p'  and  p"  exhibits  a  difference  of  -f  1".04  in  &',  and  of  —  0".06  in 
6".  Hence,  the  position  of  the  plane  of  the  orbit  as  determined  from 
the  extreme  places  very  nearly  satisfies  the  intermediate  latitudes. 

If  we  compute  the  remaining  elements  by  means  of  these  values 
of  r,  rf",  and  u,  u"r,  the  separate  results  are  : 

log  tan  G0  =  8.0522282n,  log  m0  =  9.7179026, 

log  sj  =  0.2917731,  log  x0  =  8.9608397, 
log  j9  =  0.3712405,                   I  (E"  —  E)  =  17°  35'  42".12, 

log  (a  cos  ?)  =  0.3796884,  log  cos  <p  =  9.9915521, 

w  =197°  37'  47".72,  log  e  =  9.2907906, 

<P  =    11    15  52  .46,  log  cos  <p  =  9.9915520, 

log  a  =  0.3881365,  log  ,u  =  2.9678019, 

E  =  329°  1  1'  47".24,  E'"  =  39°  34'  35".70, 

JW=334    55  40  .46,  Jf'"  =  32    26  45  .49, 

Jf0=     1    29  40  .36,  3f0=    1    29  40  .37. 

Hence,  the  elements  are  as  follows  : 

Epoch  =  1864  Jan.  1.0  Greenwich  mean  time. 
M=     1°  29'  40".36 
*=   44    20  32  .95  1  Ecli  tic  and  Mean 


P  =    11    15  52  .46 

log  a  =  0.3881365 

M  =  928".5427. 

It  appears,  therefore,  that  the  principal  effect  of  neglecting  the 
extreme  latitudes  in  the  determination  of  an  orbit  from  four  oiser 
vations  is  on  the  inclination  of  the  orbit  and  on  the  longitude  of  the 
ascending  node,  the  other  elements  being  very  slightly  changed.  The 
elements  thus  derived  represent  the  extreme  places  exactly,  and  if 
we  compute  the  second  and  third  places  directly  from  these  elements, 
v\*e  obtain 

M'  =  355°  33'  43".88,  M  "  =   9°  44'  53".73, 

^'=354    29  12  .93,  E"  =12      5  34  .81, 

vr    =  353    16  59  .07,  v"    =  14    42  45  .96. 


NUMERICAL   EXAMPLE.  80S 

log  /  =  0.2942366,  log  r"  =  0.2960826, 

u'  =      190°  54'  46".79,  u"  =      212°  20'  33".68, 

l'=        37    35  27  .75,  7'=        58    58  16  .50, 

V  =  —     0    52  21  .25,  b"  =  —     2    27  59  .06, 

/=        10    14  17  .35,  A"  =        29    53  21  .99, 

0'  =  —     1    15  47  .67,  $'  =  —     2    29  57  .62, 

log//  =  0.1334634,  log  p"=  0.2899122. 

Ilenec,  the  residuals  for  the  second  and  third  places  of  the  planet 

are  — 

Comp.  —  Obs. 

AA'  =  —  0".22,  *p  =  -f  1".53, 

A/I"  =      0  .00,  A0"  =  —  0  .06  ; 

nnd  the  elements  very  nearly  represent  the  four  normal  places.  Since 
the  interval  between  the  extreme  places  is  223  days,  these  elements 
must  represent,  within  the  limits  of  the  errors  of  observation,  the 
entire  series  of  observations  on  which  the  normals  are  based.  It 
may  be  observed,  also,  that  the  successive  approximations,  in  the 
case  of  intervals  which  are  very  large,  do  not  converge  with  the 
same  degree  of  rapidity  as  when  the  intervals  are  small,  and  that  in 
such  cases  the  numerical  calculation  is  very  much  abbreviated  by  the 
determination,  in  the  first  instance,  of  the  assumed  values  of  P',  Pn  ', 
§',  and  §"  by  means  of  approximate  elements  already  known.  For 
the  first  determination  of  an  unknown  orbit,  the  intervals  will  gene- 
rally be  so  small  that  the  first  assumed  values  of  these  quantities,  as 
determined  by  the  equations 


will  not  differ  much  from  the  correct  values,  and  two  or  three 
hypotheses,  or  even  less,  will  be  sufficient.  But  when  the  intervals 
are  large,  and  especially  if  the  eccentricity  is  also  considerable,  several 
hypotheses  may  be  required,  the  last  of  which  will  be  facilitated  by 
using  the  equations  (82)4. 

The  application  of  the  formulae  for  the  determination  of  an  orbit 
from  four  observations,  is  not  confined  to  orbits  whose  inclination  to 
the  ecliptic  is  very  small,  corresponding  to  the  cases  in  which  the 
method  of  finding  the  elements  by  means  of  three  observations  fails, 


310  THEORETICAL    ASTRONOMY. 

or  at  least  becomes  very  uncertain.  On  the  contrary,  these  formulas 
apply  equally  well  in  the  case  of  orbits  of  any  inclination  whatever, 
and  since  the  labor  of  computing  an  orbit  from  four  observations 
does  not  much  exceed  that  required  when  only  three  observed  places 
are  used,  while  the  results  must  evidently  be  more  approximate,  it 
will  be  expedient,  in  very  many  cases,  to  use  the  formulae  given  in 
this  chapter  both  for  the  first  approximation  to  an  unknown  orbit 
iud  for  the  subsequent  determination  from  more  complete  data. 


CIECULAR   OKBIT.  311 


CHAPTER  VI. 

IHVESTJGATION  OF  VARIOUS  FORMULAE  FOR  THE  CORRECTION  CF  THE  APPROXIMATE 
ELEMENTS   OF   THE   ORBIT   OF   A   HEAVENLY   BODY. 

103.  IN  the  case  of  the  discovery  of  a  planet,  it  is  often  conve- 
nient, before  sufficient  data  have  been  obtained  for  the  determination 
of  elliptic  elements,  to  compute  a  system  of  circular  elements,  an 
ephemeris  computed  from  these  being  sufficient  to  follow  the  planet 
for  a  brief  period,  and  to  identify  the  comparison  stars  used  in  dif- 
ferential observations.  For  this  purpose,  only  two  observed  places 
are  required,  there  being  but  four  elements  to  be  determined,  namely, 
&,  i,  a,  and,  for  any  instant,  the  longitude  in  the  orbit.  As  soon  as 
a  has  been  found,  the  geocentric  distances  of  the  planet  for  the 
instants  of  observation  may  be  obtained  by  means  of  the  formulae 


A  =Rcos*   -f-  I/a2  — 


J"  =  R"  cos  4"  +  Va?  —  #'«  sin2  4", 

the  values  of  $  and  fyf  being  computed  from  the  equations  (42)3  and 
(43)3.     For  convenient  logarithmic  calculation,  we  may  first  find  z 


.     „  f^ 

sin  z  =  -  ,  sin  z1'  =  -  —  ,  (2) 

a  a 

since  the  formulae  will  generally  be  required  for  cases  such  that  thes't 
angles  may  be  obtained  with  sufficient  accuracy  by  means  of  their 
sines.  Then  we  have 

sin  (*"*")  - 

~cos/9' 


from  which  to  find  p  and  p".     These  having  been  found,  we  have 

tan  (7       Crta        p  Sln  (A  ~  G) 

-' 


.    , 
smb  =  -     -Z-t 

a 


for  the  determination  of  I  and  6,  and  similarly  for  I"  and  6".     The 


312  THEOKETICAL    ASTRONOMY. 

inclination  of  the  orbit  and  the  longitude  of  the  ascending  node  are 
then  found  by  means  of  the  formulae  (75)3,  and  the  arguments  of  the 
latitude  by  means  of  (77)3.  Since  u"  —  u  is  the  distance  on  the  celes- 
tial sphere  between  two  points  of  which  the  heliocentric  spherical 
co-ordinates  are  19  b,  and  l"9  b/f,  we  have,  also,  the  equations 

sin  (u"  —  u)  sin  B  =  cos  b"  sin  (I"  —  I), 

sin  (u"  —  u)  cos  B  =  cosb  sin  b"  —  sin  b  cos  b"  cos  (I"  —  l\ 

cos  (u"  —  u)  =  sin  b  sin  b"  -f  cos  b  cos  b"  cos  (I"  —  I), 

for  the  determination  of  u" — u,  the  angle  opposite  the  side  90°  —  b" 
of  the  spherical  triangle  being  denoted  by  B.  The  solution  of  these 
equations  is  facilitated  by  the  introduction  of  auxiliary  angles,  as 
already  illustrated  for  similar  cases. 

In  a  circular  orbit,  the  eccentricity  being  equal  to  zero,  u"  —  u 
expresses  the  mean  motion  of  the  planet  during  the  interval  t" —  t, 
and  we  must  also  have 

t"-«  =  T-  («"-«)'  (6) 

the  value  of  k  being  expressed  in  seconds  of  arc,  or  log  k  =  3.5500066. 
These  formulae  will  be  applied  only  when  the  interval  t" — t  is 
small,  and  for  the  case  of  the  asteroid  planets  we  may  first  assume 

a  =  2.7, 

which  is  about  the  average  mean  distance  of  the  group.  With  this 
we  compute  p  and  pff  by  means  of  the  equations  (2)  and  (3),  and  the 
corresponding  heliocentric  places  by  means  of  (4).  If  the  inclination 
is  small,  u"  —  u  will  differ  very  little  from  I"  —  I.  Therefore,  in  the 
first  approximation,  when  the  heliocentric  longitudes  have  been  found, 
the  corresponding  value  of  t" —  t  may  be  obtained  from  equation  (5), 
writing  I"  —  I  in  place  of  u"  —  u.  If  this  comes  out  less  than  the 
actual  interval  between  the  times  of  observation,  we  infer  that  the 
assumed  value  of  a  is  too  small ;  but  if  it  comes  out  greater,  the 
assumed  value  of  a  is  too  large.  The  value  to  be  used  in  a  repetition 
of  the  calculation  may  be  computed  from  the  expression 

log  a  =  §  (log  (t"  -  0  +  log  k  -  log  (u"  -  u}), 

the  difference  u" —  u  being  expressed  in  seconds  of  arc.  With  this 
we  recompute  p,  prt9  19  and  I",  and  find  also  6,  6",  Q>9  i,  u,  and  u". 
Then,  if  the  value  of  a  computed  from  the  last  result  for  u" — u 
differs  from  the  last  assumed  value,  a  further  repetition  of  the  calcu- 


CIKCULAB   CEBIT.  313 

latiori  becomes  necessary.  But  when  three  successive  approximate 
values  of  a  have  been  found,  the  correct  value  may  be  readily  inter- 
polated according  to  the  process  already  illustrated  for  similar  cases. 

As  soon  as  the  value  of  a  has  been  obtained  which  completely 
satisfies  equation  (5),  this  result  and  the  corresponding  values  of  &, 
i,  and  the  argument  of  the  latitude  for  a  fixed  epoch,  complete  the 
system  of  circular  elements  which  will  exactly  satisfy  the  two  observed 
places.  If  we  denote  by  u0  the  argument  of  the  latitude  for  the  epoch 
T,  we  shall  have,  for  any  instant  t, 


u  being  the  mean  or  actual  daily  motion  computed  from 

Jc 


The  value  of  u  thus  found,  and  r  =  a,  substituted  in  the  formulse  for 
computing  the  places  of  a  heavenly  body,  will  furnish  the  approxi- 
mate ephemeris  required. 

The  corrections  for  parallax  and  aberration  are  neglected  in  the 
first  determination  of  circular  elements  ;  but  as  soon  as  these  approxi- 
mate elements  have  been  derived,  the  geocentric  distances  may  be 
computed  to  a  degree  of  accuracy  sufficient  for  applying  these  cor- 
rections directly  to  the  observed  places,  preparatory  to  the  determi- 
nation of  elliptic  elements.  The  assumption  of  rf  =  a  will  also  be 
sufficient  to  take  into  account  the  term  of  the  second  order  in  the  first 
assumed  value  of  P,  according  to  the  first  of  equations  (98)4. 

104.  When  approximate  elements  of  the  orbit  of  a  heavenly  body 
have  been  determined,  and  it  is  desired  to  correct  them  so  as  to  satisfy 
as  nearly  as  possible  a  series  of  observations  including  a  much  longer 
interval  of  time  than  in  the  case  of  the  observations  used  in  finding 
these  approximate  elements,  a  variety  of  -methods  may  be  applied. 
For  a  very  long  series  of  observations,  the  approximate  elements 
being  such  that  the  squares  of  the  corrections  which  must  be  applied 
to  them  may  be  neglected,  the  most  complete  method  is  to  form  the 
equations  for  the  variations  of  any  two  spherical  co-ordinates  which 
fix  the  place  of  the  body  in  terms  of  the  variations  of  the  six  ele- 
ments of  the  orbit;  and  the  differences  between  the  computed  places 
for  different  dates  and  the  corresponding  observed  places  thus  furnish 
equations  of  condition,  the  solution  of  which  gives  the  corrections  to 
be  applied  to  the  elements  But  when  the  observations  do  not  in- 


314  THEORETICAL   ASTRONOMY. 

elude  a  very  long  interval  of  time,  instead  of  forming  the  equations 
for  the  variations  of  the  geocentric  places  in  terms  of  the  variations 
of  the  elements  of  the  orbit,  it  will  be  more  convenient  to  form  the 
equations  for  these  variations  in  terms  of  quantities,  less  in  number, 
from  which  the  elements  themselves  are  readily  obtained.  If  no  as- 
sumption is  made  in  regard  to  the  form  of  the  orbit,  the  quantities 
which  present  the  least  difficulties  in  the  numerical  calculation  are 
the  geocentric  distances  of  the  body  for  the  dates  of  the  extreme 
observations,  or  at  least  for  the  dates  of  those  which  are  best  adapted 
to  the  determination  of  the  elements.  As  soon  as  these  distances  are 
accurately  known,  the  two  corresponding  complete  observations  are 
sufficient  to  determine  all  the  elements  of  the  orbit. 

The  approximate  elements  enable  us  to  assume,  for  the  dates  t  and 
tn,  the  values  of  A  and  J";  and  the  elements  computed  from  these 
by  means  of  the  data  furnished  by  observation,  will  exactly  represent 
the  two  observed  places  employed.  Further,  the  elements  may  be 
supposed  to  be  already  known  to  such  a  degree  of  approximation  that 
the  squares  and  products  of  the  corrections  to  be  applied  to  the 
assumed  values  of  A  and  A"  may  be  neglected,  so  that  we  shall  have, 
for  any  date, 

..  da,  .     da 

COS  S  Aa  =  COS  d  — -  A  J  -j-  COS  d  -=---rr  A  A   , 

d  d  fR\ 

d*  d3 

**  =  dA^  +  ^FAJ' 

If,  therefore,  we  compare  the  elements  computed  from  A  and  A"  with 
any  number  of  additional  or  intermediate  observed  places,  each  ob- 
served spherical  co-ordinate  will  furnish  an  equation  of  condition  for 
the  correction  of  the  assumed  distances.  But  in  order  that  the  equa- 
tions (6)  may  be  applied,  the  numerical  values  of  the  partial  differen- 
tial coefficients  of  a  and  d  with  respect  to  A  and  d"  must  be  found. 
Ordinarily,  the  best  method  of  effecting  the  determination  of  these  is 
to  compute  three  systems  of  elements,  the  first  from  A  and  A",  the 
second  from  A  +  D  and  J",  and  the  third  from  A  and  A"  -f  D",  D 
and  D"  being  small  increments  assigned  to  A  and  A"  respectively. 
]  f  now,  for  any  date  t'y  we  compute  a/  and  df  from  each  system  of 
elements  thus  obtained,  we  may  find  the  values  of  the  differential 
coefficients  sought.  Thus,  let  the  spherical  co-ordinates  for  the  time 
tf  computed  from  the  first  system  be  denoted  by  a'  and  3f ;  those 
computed  from  the  second  system  of  elements,  by  a/  -f  a  sec  3f  and 
6'  +  d;  and  those  from  the  third  system,  by  a'+  a!'  sec  d'  and  d'-\-  d". 
Then  we  shall  have 


VARIATION   OF   TWO   GEOCENTRIC   DISTANCES.  315 

..  da!         a  dd        d 


,,a__  -.. 

dA"~  D'"  dA"~  D"' 


and  the  equations  (6)  give 


COS  dr  Aa'  =  ~  A  A  -f  -^77  A  J", 

d      d"  (8) 


In  the  same  manner,  computing  the  places  for  various  dates,  for 
which  observed  places  are  given,  by  means  of  each  of  the  three  systems 
of  elements,  the  equations  for  the  correction  of  A  and  A"  ,  as  deter- 
mined by  each  of  the  additional  observations  employed,  may  be 
formed. 

105.  For  the  purpose  of  illustrating  the  application  of  this  method, 
let  us  suppose  that  three  observed  places  are  given,  referred  to  the 
ecliptic  as  the  fundamental  plane,  and  that  the  corrections  for  parallax, 
aberration,  precession,  and  nutation  have  all  been  duly  applied.  By 
means  of  the  approximate  elements  already  known,  we  compute  the 
values  of  A  and  A"  for  the  extreme  places,  and  from  these  the  helio- 
centric places  are  obtained  by  means  of  the  equations  (71)3  and  (72)3, 
writing  A  cos  /9  and  A"  cos  ft"  in  place  of  p  and  p".  The  values  of 
&,  i,  u,  and  u"  will  be  obtained  by  means  of  the  formulae  (76)3  and 
(77)3;  and  from  r,  r"  and  u"  —  u  the  remaining  elements  of  the 
orbit  are  determined  as  already  illustrated.  The  first  system  of  ele- 
ments is  thus  obtained.  Then  we  assign  an  increment  to  J,  which 
we  denote  by  D,  and  with  the  geocentric  distances  A  -f-  D  and  A" 
we  compute  in  precisely  the  same  manner  a  second  system  of  ele- 
ments. Next,  we  assign  to  A"  an  increment  D",  and  from  A  and 
A"  +  D"  a  third  system  of  elements  is  derived.  Let  the  geocentric 
longitude  and  latitude  for  the  date  of  the  middle  observation  com- 
puted from  the  first  system  of  elements  be  designated,  respectively, 
by  V  and  /9/  ;  from,  the  second  system  of  elements,  by  V  and  /?2'  ; 
and  from  the  third  system,  by  ^/  and  /93'.  Then  from 


It  (  -\   I  i  l\  of  fjff  O  f  Of  \**  s 

we  compute  a,  a",  d,  and  d" ',  and  by  means  of  these  and  the  values 
of  D  and  D"  we  form  the  equations 


316  THEORETICAL     ASTRONOMY. 


=  CM 


for  the  determination  of  the  corrections  to  be  applied  to  the  f.rst 
assumed  values  of  A  and  An ',  by  means  of  the  differences  between 
observation  and  computation.  The  observed  longitude  and  latitude 
being  denoted  by  X  and  /?',  respectively,  we  shall  have 

COS  f  A/  =  (/  —  A/)  COS  /?', 

±ff=ff  —  B'  (11) 

for  finding  the  values  of  the  second  members  of  the  equations  (10), 
and  then  by  elimination  we  obtain  the  values  of  the  corrections  A  J 
and  AJ"  to  be  applied  to  the  assumed  values  of  the  distances. 
Finally,  we  compute  a  fourth  system  of  elements  corresponding  to 
the  geocentric  distances  A  -f  A  A  and  A"  -f  A  A"  either  directly  from 
these  values,  or  by  interpolation  from  the  three  systems  of  elements 
already  obtained ;  and,  if  the  first  assumption  is  not  considerably  in 
error,  these  elements  will  exactly  represent  the  middle  place.  It 
should  be  observed,  however,  that  if  the  second  system  of  elements 
represents  the  middle  place  better  than  the  first  system,  A/  and  /?2r 
should  be  used  instead  of  ^/  and  /9/  in  the  equations  (11),  and,  in 
this  case,  the  final  system  of  elements  must  be  computed  with  the 
distances  A  -f-  D  +  A  A  and  A"  +  A  A".  Similarly,  if  the  middles 
place  is  best  represented  by  the  third  system  of  elements,  the  cor- 
rections will  be  obtained  for  the  distances  used  in  the  third  hy- 
pothesis. 

If  the  computation  of  the  middle  place  by  means  of  the  final  ele- 
ments still  exhibits  residuals,  on  account  of  the  neglected  terms  of 
the  second  order,  a  repetition  of  the  calculation  of  the  corrections 
A/f  and  AJ",  using  these  residuals  for  the  values  of  the  second 
members  of  the  equations  (10),  will  furnish  the  values  of  the  dis- 
tances for  the  extreme  places  with  all  the  precision  desired.  The 
increments  D  and  D"  to  be  assigned  successively  to  the  first  assumed 
values  of  A  and  A"  may,  without  difficulty,  be  so  taken  that  the 
true  elements  shall  differ  but  little  from  one  of  the  three  systems 
computed ;  and  in  all  the  formulae  it  will  be  convenient  to  use,  in- 
stead of  the  geocentric  distances  themselves,  the  logarithms  of  these 
distances,  and  to  express  the  variations  of  these  quantities  in  units 
of  the  last  decimal  place  of  the  logarithms. 

These  formulae  will  generally  be  applied   for  the  correction  of 


VARIATION   OF   T\VO   GEOCENTRIC   DISTANCES.  317 

approximate  elements  by  means  of  several  observed  places,  which 
may  be  either  single  observations  or  normal  places,  each  derived  from 
several  observations,  and  the  two  places  selected  for  the  computation 
of  the  elements  from  A  and  A"  should  not  only  be  the  most  accurate 
possible,  but  they  should  also  be  such  that  the  resulting  elements  are 
not  too  much  affected  by  small  errors  in  these  geocentric  places. 
They  should  moreover  be  as  distant  from  each  other  as  possible,  the 
other  considerations  not  being  overlooked.  When  the  three  systems 
of  elements  have  been  computed,  each  of  the  remaining  observed 
places  will  furnish  two  equations  of  condition,  according  to  equations 
(10),  for  the  determination  of  the  corrections  to  be  applied  to  the 
assumed  values  of  the  geocentric  distances ;  and,  since  the  number 
of  equations  will  thus  exceed  the  number  of  unknown  quantities, 
the  entire  group  must  be  combined  according  to  the  method  of  least 
squares.  Thus,  we  multiply  each  equation  by  the  coefficient  of  AJ 
in  that  equation,  taken  with  its  proper  algebraic  sign,  and  the  sum 
of  all  the  equations  thus  formed  gives  one  of  the  final  equations 
required.  Then  we  multiply  each  equation  by  the  coefficient  of  AJ" 
in  that  equation,  taken  also  with  its  proper  algebraic  sign,  and  the 
sum  of  all  these  gives  the  second  equation  required.  From  these 
two  final  equations,  by  elimination,  the  most  probable  values  of  AJ 
and  AZ/;/  will  be  obtained ;  and  a  system  of  elements  computed  with 
the  distances  thus  corrected  will  exactly  represent  the  two  funda- 
mental places  selected,  while  the  sum  of  the  squares  of  the  residuals 
for  the  other  places  will  be  a  minimum.  The  observations  are  thus 
supposed  to  be  equally  good;  but  if  certain  observed  places  are 
entitled  to  greater  influence  than  the  others,  the  relative  precision 
of  these  places  must  be  taken  into  account  in  the  combination  of  the 
equations  of  condition,  the  process  for  which  will  be  fully  explained 
in  the  next  chapter. 

When  a  number  of  observed  places  are  to  be  used  for  the  correction 
of  the  approximate  elements  of  the  orbit  of  a  planet  or  comet,  it  w*!ll 
be  most  convenient  to  adopt  the  equator  as  the  fundamental  plane. 
In  this  case  the  heliocentric  places  will  be  computed  from  the  assumed 
values  of  A  and  J",  and  the  corresponding  geocentric  right  ascensions 
and  declinations  by  means  of  the  formulae  (106)3  and  (107)3;  and  the 
position  of  the  plane  of  the  orbit  as  determined  from  these  by  means 
of  the  equations  (76)3  will  be  referred  to  the  equator  as  the  funda- 
mental plane.  The  formation  of  the  equations  of  condition  for  the 
corrections  AJ  and  A  A"  to  be  applied  to  the  assumed  values  of  the 
distances  will  then  be  effected  precisely  as  in  the  case  of  I  and  /?,  the 


318  THEORETICAL   ASTRONOMY. 

necessary  changes  being  made  in  the  notation.  In  a  similar  manner, 
the  calculation  may  be  effected  for  any  other  fundamental  plane  which 
may  be  adopted. 

It  should  be  observed,  further,  that  when  the  ecliptic  is  taken  as 
the  fundamental  plane,  the  geocentric  latitudes  should  be  corrected 
by  means  of  the  equation  (6)4,  in  order  that  the  latitudes  of  the  sun 
phall  vanish,  otherwise,  for  strict  accuracy,  the  heliocentric  places 
must  be  determined  from  A  and  A"  in  accordance  with  the  equations 
(39):- 

106.  The  partial  differential  coefficients  of  the  two  spherical  co- 
ordinates with  respect  to  A  and  A"  may  be  computed  directly  by 
means  of  differential  formula;  but,  except  for  special  cases,  the 
numerical  calculation  is  less  expeditious  than  in  the  case  of  the  indi- 
rect method,  while  the  liability  of  error  is  much  greater.  If  we 
adopt  the  plane  of  the  orbit  as  determined  by  the  approximate  values 
of  A  and  A"  as  the  fundamental  plane,  and  introduce  /  as  one  of  the 
elements  of  the  orbit,  as  in  the  equations  (72)2,  the  variation  of  the 
geocentric  longitude  6  measured  in  this  plane,  neglecting  terms  of  the 
second  order,  depends  on  only  four  elements;  and  in  this  case  the 
differential  formulae  may  be  applied  with  facility.  Thus,  if  we  ex- 
press r  and  v  in  terms  of  the  elements  ^,  MQ)  and  /*,  we  shall  have 


and 
or 

In  like 

dr 
dA 

dv 
dA 

_dr 
dtp 

dv 

dtp 

dr 

dM0 

dr 

h  d,u 

dv 

dp. 

dp. 
'  dA 

'dA 
•£-> 

dv 

dA 

'  dA' 
dp. 

dx 

dMQ 
d<p 

dA 
dv 

1      dfJL 

dM0 

dv 

dA 
manner,  we 

dA 
have 

d(f> 

'dA 

dM0 

dA 

1    dp. 

__  _ 

d?  '  dA        dM0  '   dA   "     dfi  '  dA' 


dA          ~  d<p    dA       dM0      dA         d.a    dA    '   dA' 

dr    d(v+fi    dr"  cW-t-/) 

As  soon  as  the  values  of  -=-p  —  -T-.  —  >    TT>  an(i  -  n  --   are 

dA          dA          dA  dA 

known,  the  equations  necessary  for  finding  the  differential  coefficients 
of  the  elements  £,  <p,  Mw  and  /z  with  respect  to  A  are  thus  provided. 
In  the  case  under  consideration,  when  an  increment  is  assigned  to  A, 


VARIATION   OF   TWO   GEOCENTRIC   DISTANCES.  319 

the  value  of  A"  remaining  unchanged,  r"  and  v"  -f  /  are  not  changed, 
and  hence 

*1  =  0.          ^+i>  =  o. 

dA  aA 

To  find  -7-7-  and  -  7  .      ,  from  the  equations 
dA  dA 

A  cos  ^  cos  0  =  x  -f-  X, 
A  cos  >?  sin  0  =  y  -{-  F", 

in  which  ^  is  the  geocentric  latitude  in  reference  to  the  plane  of  the 
orbit  computed  from  A  and  A"  as  the  fundamental  plane,  and  X,  Y 
the  geocentric  co-ordinates  of  the  sun  referred  to  the  same  plane,  we 

get 

dx  =  cos  7]  cos  6  dA, 
dy  =  cos  T?  sin  6  dA, 

or,  substituting  for  dx  and  dy  their  values  given  by  (73)2, 

cos  t]  cos  6  dA  =  cos  u  dr  —  r  sin  u  d  (v  -f-  /), 
cos  r)  sin  O  d  A  =  sin  udr  -\-  r  cos  it  c?  (v  -f~  /). 

Eliminating,  successively,  c?  (v  +  #)  and  c?r,  we  get 

dr  ,n        x 

-—  =  cos  ^  cos  (6  —  u), 

d 

I  .    ,0        . 

=  -  cos  ^  sm  (6  —  u). 


7  - 

dA  r 

Therefore,  we  shall  have 
dy    ,     dv     dv         dv 


dip     dA    '    dMQ      dA          dp.     dA  ,_ 

dX_      _dif_    d?_        d^_    dM^,dv^_   ^_0 
dA  H"          '  dA  +  ^jfo      jj    '  '    ^    '  ^j 

_dr^_    ^__0 
'       ~ 


^     dA       dif0      dA          dfji     dA 

and  if  we  compute  the  numerical  values  of  the  differential  coefficients 
of  r,  r",  v,  and  vtr  with  respect  to  the  elements  <p,  M0,  and  //,  these 
equations  will  furnish,  by  elimination,  the  values  of  the  four  un- 

d%   d<p    dMQ        ,  dfj. 
known  quantities  ^  ^  -^,  and  ^j 

Tn  precisely  the  same  manner  we  derive  the  following  equations 


320  THEORETICAL  "ASTRONOMY. 

for  the  determination  of  the  partial  differential  coefficients  of  these 
elements  with  respect  to  A" : — 

dx     ,    dv      d(p          dv      dMQ        dv      dfj.     _  ~ 
dA*  +  ~dj'  ~d^  +  dMl '  JA77  "*"  ~cfc  '  ~dAFr  ~=    ' 


dr      dtp          dr      dM^        dr      dp.     _ 
~dj "dA*  +  dfio  '  ~dA*  nh  ~dfl'~dlf"      ' 

d/         dt/;     efr         <fo"     (Of0    ,   dv"      dfi          \  „    .     ,  _„  _  (  ^ 

^  +  ^'^^dM~'~dA*'~]r  dp.'  dA"-~^r™*r>    Sm(t          U  }' 


dr"     d<?         dr"     dM0    ,   dr" 


dA"    r  dft     dA" 

Since  the  geocentric  latitude  ^  is  affected  chiefly  by  a  change  of  the 
position  of  the  plane  of  the  orbit,  while  the  variation  of  the  longitude 
6  is  independent  of  &  and  i  when  the  squares  and  products  of  the 
variations  of  the  elements  are  neglected,  if  we  determine  the  elements 
which  exactly  represent  the  places  to  which  A  and  A"  belong,  as  well 
as  the  longitudes  for  two  additional  places,  or,  if  we  determine  those 
which  satisfy  the  two  fundamental  places  and  the  longitudes  for  any 
number  of  additional  observed  places,  so  that  the  sum  of  the  squares 
of  their  residuals  shall  be  a  minimum,  the  results  thus  obtained  will 
very  nearly  satisfy  the  several  latitudes. 

Let  0'  denote  the  geocentric  longitude  of  the  body,  referred  to  the 
plane  of  the  orbit  computed  from  A  and  A"  as  the  fundamental  plane, 
for  the  date  tr  of  any  one  of  the  observed  places  to  be  used  for  cor- 
recting these  assumed  distances.  Then,  to  find  the  partial  differential 
coefficients  of  6f  with  respect  to  A  and  A",  we  have 

,  do'  .do'     dy  .do'      dy  .  dtf     dMQ 

COS  rj  -7-7-  =  COS  TJ  -j ~-  -f  COS y  3—  •  -j-j-  +  COS  ff  -7-=-  •  — j-f 

dA  d/      dA  d<p      dA  dMQ      dA 

-f-  cos  f/  -=—  •    T  .  »  ,-K, 

dp.      dA  (15) 

do'  _fdO'     dy,     ,       ^  ,dtf_     d<p     ,   f  _,  dtf_ 

,  do'     dn 


and  by  means  of  the  results  thus  derived,  we  form  the  equation 

COS  rl  *tf  =  COS  r!^  *A  +  COS  V  ~  A  J".  (16) 

A  fourth  observed  place  will  furnish,  in  the  same  manner,  the  addi- 
tional equation  required  for  finding  A  J  and  A  A".     If  more  than  two 


VARIATION   OF   TWO  .GEOCENTRIC   DISTANCES.  321 

observations  are  used  in  addition  to  the  fundamental  places  on  which 
the  assumed  elements  as  derived  from  A  and  A"  are  based,  the  several 
longitudes  will  furnish  each  an  equation  of  condition,  and  the  most 
probable  values  of  AZ/  and  &A"  will  be  obtained  by  combining  the 
entire  group  of  equations  of  condition  according  to  the  method  of 
least  squares. 

107.  In  the  actual  application  of  these  formulae  to  the  correction 
of  the  approximate  elements,  after  all  the  preliminary  corrections 
have  been  applied  to  the  data,  we  select  the  proper  observed  places 
for  determining  the  elements  from  the  corresponding  assumed  dis- 
tances A  and  An ',  according  to  the  conditions  which  have  already  been 
stated,  and  from  these  we  derive  the  six  elements  of  the  orbit.  Since 
the  data  furnished  directly  by  observation  are  the  right  ascensions 
and  the  declinations  of  the  body,  the  elements  will  be  derived  in 
reference  to  the  equator  as  the  plane  to  which  the  inclination  and  the 
longitude  of  the  ascending  node  belong.  These  elements  will  exactly 
represent  the  two  fundamental  places,  and,  if  the  assumed  distances 
A  and  A"  are  not  much  in  error,  they  will  also  very  nearly  satisfy 
the  remaining  places. 

We  now  adopt  as  the  fundamental  plane  the  plane  of  the  approxi- 
mate orbit  thus  determined,  and  by  means  of  the  equations  (83)2  and 
(85)2,  or  by  means  of  (87)2,  writing  a,  d,  & ',  and  i'  in  place  of  A,  /9, 
Q>,  and  •£,  respectively,  we  compute  the  values  of  0,  37,  and  f  for  the 
dates  of  the  several  places  to  be  employed.  Then  the  residuals  for 
each  of  the  observed  places  are  found  from  the  formulae 

cos  f)  A0  =  sin  Y  A<5  -f-  cos  f  cos  8  Aa, 
Aiy  =  cos  Y  A#  —  sin  f  cos  d  Aa, 

the  values  of  Aa  and  A£  for  each  place  being  found  by  subtracting 
from  the  observed  right  ascension  and  declination,  respectively,  the 
right  ascension  and  declination  computed  by  means  of  the  elements 
derived  from  A  and  A".  The  values  of  09  f],  and  f  being  required 
only  for  finding  cos  y  A#,  A/7,  and  the  differential  coefficients  of  d  and 
ry,  with  respect  to  the  elements  of  the  orbit,  need  not  be  determined 
with  great  accuracy. 

Next,  we  compute  -^  and  -  k    ,  .        from  equations  (12),  and  from 

,+ n^     .1          i  „  dr    dr"    dv    dv"    dr     0        ,  /»     i  •  i 

(16),  the  values  of  — ,  — ,  — ,  -=— ,  -r^,  &c,,  by  means  of  which, 
d<?    d<p     d(p    d(p    dMQ 

using  the  value  of  u  in  reference  to  the  equator,  we  form  the  equa- 
tions (13).  The  accent  is  added  to  /  to  indicate  that  it  refers  to  the 

21 


322  THEORETICAL    ASTRONOMY. 

equator  as  the  plane  for  defining  the  elements.  Thus  we  obtain  four 
equations,  from  which,  by  elimination,  the  values  of  the  differential 
coefficients  of  *£r,  <p,  Jf0,  and  [JL  with  respect  to  J  may  be  obtained. 
In  the  numerical  solution,  by  subtracting  the  third  equation  from 

dyf 
the  first,  the  unknown  quantity  -^r  is  immediately  eliminated,  so  that 

we  have  three  equations  to  find  the  three  unknown  quantities  —  , 
--TJ,  and  -jj  These  having  been  found,  ~  may  be  obtained  from 

the  first  or  from  the  third  equation. 

In  the  same  manner  we  form  the  equations  (14),  and  thence  derive 

the  values  of  -y^p  -7-^,  -r-£-,  and   -7177-     Then,  by  means  of  the  for- 
ofij     CL^J     d^j  U--J 

mulse  (76)2,  (78)2,  and  (79)2,  we  compute  for  the  date  of  each  place 
to  be  employed  in  correcting  the  assumed  distances  the  values  of 

cos?/  -p,  COST/  —,,  &c.,  and  hence  from  (15)  the  values  of  cos^'-r^ 
and  cos  r/  ——•  The  results  thus  obtained,  together  with  the  residuals 

computed  by  means  of  the  equations  (17),  enable  us  to  form,  accord- 
ing to  (16),  the  equations  of  condition  for  finding  the  values  of  the 
corrections  AJ  and  &A"  .  The  solution  of  all  the  equations  thus 
formed,  according  to  the  method  of  least  squares,  will  give  the  most 
probable  values  of  these  quantities,  and  the  system  of  elements  which 
corresponds  to  the  distances  thus  corrected  will  very  nearly  satisfy 
the  entire  series  of  observations.  Since  the  values  of  cos  r/  A0'  are 
expressed  in  seconds  of  arc,  the  resulting  values  of  A  A  and  A  A"  will 
also  be  expressed  in  seconds  of  arc  in  a  circle  whose  radius  is  equal 
to  the  mean  distance  of  the  earth  from  the  sun.  To  express  them  in 
parts  of  the  unit  of  space,  we  must  divide  their  values  in  seconds  of 
arc  by  206264.8. 

The  corrections  to  be  applied  to  the  elements  computed  from  A  and 
J",  in  order  to  satisfy  the  corrected  values  A  -f-  A  J  and  A"  -f  A  A", 
may  be  computed  by  means  of  the  partial  differential  coefficients 
already  derived.  Thus,  in  the  case  of  j£r,  we  have 


from  which  to  find  A/'  ;  and  in  a  similar  manner  A^>,  A!/O,  and 

may  be  obt 
we  compute 


may  be  obtained.     If,  from  the  values  of  --  -     —    and  —  -p  — 


VARIATION   OF   TWO   GEOCENTRIC   DISTANCES.  323 


and  apply  these  corrections  to  the  values  of  v  and  v"  found  from  J 
and  J",  we  obtain  the  true  anomalies  corresponding  to  the  distances 
A  -}-  A  J  and  J"  +  A  J".  The  corrections  to  be  applied  to  the  values 
of  r  and  r"  derived  from  A  and  A"  are  given  by 


If  A  J  and  AJ"  are  expressed  in  seconds  of  arc,  the  corresponding 
values  of  Ar  and  Ar"  must  be  divided  by  206264.8.  The  corrected 
results  thus  obtained  should  agree  with  the  values  of  r  and  r"  com- 
puted directly  from  the  corrected  values  of  v,  v",  p,  and  e  by  means 
of  the  polar  equation  of  the  conic  section.  Finally,  we  have 

dz  =  sin  rj  dd, 
and  similarly  for  dzff  •  and  the  last  of  equations  (73)2  gives 

r  sin  u  Ai'  —  r  cos  u  sin  i'  A  &  '         ==  sin  7  A  J, 
r"  sin  u"  Ai'  —  r"  cos  u"  sin  i'  A  ft  '  —  sin  ^  A  J", 

from  which  to  find  AIV  and  A  &  ',  u  and  u"  being  the  arguments  of 
the  latitude  in  reference  to  the  equator.  We  have  also,  according  to 
(72)* 

A  a/  =  A/  —  COS  i'  A  &  ', 
4^  =  A/  +  2  Sin1  £t*  A  &', 

from  which  to  find  the  corrections  to  be  applied  to  a)1  and  nf.  The 
elements  which  refer  to  the  equator  may  then  be  converted  into  those 
for  the  ecliptic  by  means  of  the  formula  which  may  be  derived  from 
(109)!  by  interchanging  &  and  &'  and  180°  —  i'  and  i. 

The  final  residuals  of  the  longitudes  may  be  obtained  by  substi- 
tuting the  adopted  values  of  A  J  and  A  A"  in  the  several  equations  of 
condition,  or,  which  affords  a  complete  proof  of  the  accuracy  of  the 
entire  calculation,  by  direct  calculation  from  the  corrected  elements  ; 
and  the  determination  of  the  remaining  errors  in  the  values  of  rj  will 
show  how  nearly  the  position  of  the  plane  of  the  orbit  corresponding 
to  the  corrected  distances  satisfies  the  intermediate  latitudes. 

Instead  of  <p,  Mw  and  /*,  we  may  introduce  any  other  elements 
which  determine  the  form  and  magnitude  of  the  orbit,  the  necessary 


324  THEORETICAL   ASTRONOMY. 

changes  being  made  in  the  formulae.  Thus,  if  we  use  the  elements 
T,  <?,  and  e,  these  must  be  written  in  place  of  Mw  p,  and  <py  respect- 
ively, in  the  equations  (13),  (14),  and  (15),  and  the  partial  diiferential 
coefficients  of  r,  r",  v,  and  v"  with  respect  to  these  elements  must  be 
computed  by  means  of  the  various  diiferential  formulae  which  have 
already  been  investigated.  Further,  in  all  these  cases,  the  homo- 
geneity of  the  formulae  must  be  carefully  attended  to. 

108.  The  approximate  elements  of  the  orbit  of  a  heavenly  body 
may  also  be  corrected  by  varying  the  elements  which  fix  the  position 
of  the  plane  of  the  orbit.  Thus,  if  the  observed  longitude  and  lati- 
tude and  the  values  of  &  and  i  are  given,  the  three  equations  (91)t 
will  contain  only  three  unknown  quantities,  namely,  //,  r,  and  u,  and 
the  values  of  these  may  be  found  by  elimination.  When  the  observed 
latitude  /9  is  corrected  by  means  of  the  formula  (6)4,  the  latitudes  of 
the  sun  disappear  from  these  equations,  and  if  we  multiply  the  first 
by  sin  (O  —  &)  sin  /?,  the  second  (using  only  the  upper  sign)  by 
—  cos  ( O  —  &)  sin  /9,  and  the  third  by  —  sin  (^  —  O)  cos  /9,  and  add 
the  products,  we  get 

sin /9  sin  (O —  &_) 

~  cos  i  sin  /?  cos  (O  —  &  )  —  sin  i  cos  /5  sin  (A  —  Q/ 

from  which  u  may  be  found.  If  we  multiply  the  second  of  these 
equations  by  sin  /?,  and  the  third  by  —  cos /?  sin  (A  —  &),  and  add  the 
products,  we  find 

*  /     *          *  J-    /Q       *«   /  }  /"V  \  »~  «**V*  \  / 


sin  u  (sin  i  cot  ft  sin  (A  —  &  )  —  cos  i) 

The  expression  for  r  in  terms  of  the  known  quantities  may  also  be 
found  by  combining  the  first  and  second,  or  by  combining  the  first 
and  third,  of  equations  (91)r  If  we  put 

n  cosN=  sin  /?  cos  (Q  —  &), 
n  sin  N=  cos  /?  sin  (A  —  O). 

the  formula  for  u  becomes 

tan  it  — . , T  ,  ...  tan  ( Q  —  &).  (21) 

cos  (JV-|-»J 

The  last  of  equal  ions  (91)t  shows  that  sinw  and  sin/?  must  have  the 
game  sign,  and  thus  the  quadrant  in  which  u  must  be  taken  is  deter- 
mined. Putting,  also, 


rmcos      = 

m  sin  M  =  sin  u  cot  £  sin  (A  — 


VAKIATION   OF   THE   NODE   AND   INCLINATION.  325 

we  have 

r  —  _      cos^      .  R sm  ( Q  —  &  )  (22) 

cos  (M  -\- 1)  sin  u 

When  any  other  plane  is  taken  as  the  fundamental  plane,  the 
latitude  of  the  sun  (which  will  then  refer  to  this  plane)  will  be  re- 
tained in  the  equations  (91)i  and  in  the  resulting  expressions  for  u 
and  r. 

The  value  of  u  may  also  be  obtained  by  first  computing  w  and  ^ 
by  means  of  the  equations  (42)3,  and  then,  if  z  denotes  the  angle  at 
the  planet  or  comet  between  the  earth  and  sun,  the  values  of  u  and 
2,  as  may  be  readily  seen,  will  be  determined  by  means  of  the  rela- 
tions of  the  parts  of  a  spherical  triangle  of  which  the  sides  are 
180°  —  (z  -j-  4-X  180°  -f  O  —  &,  and  u,  the  angle  opposite  to  the 
side  u  being  that  which  we  designate  by  w,  and  the  side  180°  +  O  —  Q, 
being  included  by  this  and  the  inclination  i.  Let  8=  180°  —  (z  +  -^)f 
and,  according  to  Napier's  analogies,  this  spherical  triangle  gives 

,        cos  i  (*  —  tu) 

tan  i  (£ -f  *}  = f77-7  —^  cot  i  (£  —  O), 

2  cosJCi-hw) 

0,'r,     1     /'/,'   «,^  V60/ 


from  which  8  and  u  are  readily  found.     Then  we  have 

z  =  180°  —  *—8 


sn  2 
to  find  r. 

If  we  assume  approximate  values  of  &  and  i,  as  given  by  a  system 
of  elements  already  known,  the  equations  here  given  enable  us  to  find 
r,  u,  r"  ',  and  u"  from  ^,  /9  and  X",  ft",  corresponding  to  the  dates  t 
and  t"  of  the  fundamental  places  selected,  and  from  these  results  for 
two  radii-  vectores  and  arguments  of  the  latitude,  the  remaining 
elements  may  be  derived.  From  these  the  geocentric  place  of  the 
body  may  be  found  for  the  date  tr  of  any  intermediate  or  additional 
observed  place,  and  the  difference  between  the  computed  and  the 
observed  place  will  indicate  the  degree  of  precision  of  the  assumed 
values  of  &  and  i.  Then  we  assign  to  &  the  increment  £&,  i 
remaining  unchanged,  and  compute  a  second  system  of  elements,  and 
from  these  the  geocentric  place  for  the  time  tr.  We  also  compute  a 
third  system  from  &  and  i  -j-  3i,  and  by  a  process  entirely  analogous 
to  that  already  indicated  in  the  case  of  the  variation  of  two  geocentric 


326  THEOEETICAL   ASTKONOMY. 

distances,  we  obtain  the  numerical  values  of  the  differential  coeffi- 
cients of  A'  and  ft'  with  respect  to  &  and  i.     Thus  the  equations 

COS  p  AA'  =  COS  jf  -^-  A  ft  +  COS  jf  ^  Al, 

a£g  at  ,„. 

'  C       •' 


for  finding  the  corrections  A&  and  A^  to  be  applied  to  the  assumed 
values  of  these  elements,  will  be  formed;  and  each  additional  obser- 
vation or  normal  place  will  furnish  two  equations  of  condition  for 
the  determination  of  these  corrections. 

If  the  observed  right  ascensions  and  declinations  are  used  directly 
instead  of  the  longitudes  and  latitudes,  the  elements  Q  and  i  must 
be  referred  to  the  equator  as  the  fundamental  plane,  and  the  declina- 
tions of  the  sun  will  appear  in  the  formulae  for  u  and  r  obtained  from 
the  equations  (91)^  thus  rendering  them  more  complex.  Their  deri- 
vation offers  no  difficulty,  being  similar  in  all  respects  to  that  of  the 
equations  (19)  and  (20),  and  since  they  will  be  rarely,  if  ever,  re- 
quired, it  is  not  necessary  to  give  the  process  here  in  detail.  In 
general,  the  equations  (23)  and  (24)  will  be  most  convenient  for 
finding  r  and  u  from  the  geocentric  spherical  co-ordinates  and  the 
elements  Q  and  i,  since  w,  ^,  wff,  and  ij/'  remain  unchanged  for  the 
three  hypotheses. 

When  the  equator  is  taken  as  the  fundamental  plane,  ^  is  the 
distance  between  two  points  on  the  celestial  sphere  for  which  the 
geocentric  spherical  co-ordinates  are  A,  D  and  a,  d,  those  of  the  sun 
being  denoted  by  A  and  D.  Hence  we  shall  have 

sin  4  sin  B  =  cos  d  sin  (a  —  A), 

sin  4  cos  B  =  cos  D  sin  8  —  sin  D  cos  d  cos  (a  —  A),  (26) 

cos  4»  =  sin  D  sin  d  -}-  cos  D  cos  $  cos  (a  —  A), 

from  which  to  find  ij/  and  B,  the  angle  opposite  to  the  side  90°  —  d 
of  the  spherical  triangle  being  denoted  by  B.  Let  K  denote  the 
right  ascension  of  the  ascending  node  on  the  equator  of  a  great  circle 
passing  through  the  places  of  the  sun  and  comet  or  planet  for  the 
time  tj  and  let  w0  denote  its  inclination  to  the  equator;  then  we  shall 

have 

sin  w0  cos  (A  —  -R")  =  cos  B, 

sin  WQ  sin  (A  —  K)  =  sin  B  sin  D,  (27) 

cos  WQ  —  sin  B  cos  D, 

from  which  to  find  WQ  and  K.     In  a  similar  manner,  we  may  com- 


VARIATION   OF   THE   NODE   AND   INCLINATION.  327 

pute  the  values  of  u"  —  it,  Q,  and  i  from  the  heliocentric  spherical 
co-ordinates  I,  b  and  I",  6". 
From  the  equations 


the  accents  being  added  to  distinguish  the  elements  in  reference  to 
the  equator  from  those  with  respect  to  the  ecliptic,  the  values  of  80 
and  u  (in  reference  to  the  equator)  may  be  found.  Let  SQ  denote  the 
angular  distance  between  the  place  of  the  sun  and  that  point  of  the 
equator  for  which  the  right  ascension  is  K,  and  the  equation 

cot  s0  =  cos  WQ  cot  (IT  —  A)  (29) 

gives  the  value  of  s0,  the  quadrant  in  which  it  is  situated  being  deter- 
mined by  the  condition  that  coss0  and  cos(JT  —  A)  shall  have  the 
same  sign.  Then  we  have  S  =  SQ  —  SQ,  and 


_  E  sin  4 

sin  z 
from  which  to  find  r. 


109.  In  both  the  method  of  the  variation  of  two  geocentric  dis- 
tances and  that  of  the  variation  of  &  and  i,  instead  of  using  the 
geocentric  spherical  co-ordinates  given  by  an  intermediate  observa- 
tion, in  forming  the  equations  for  the  corrections  to  be  applied  to  the 
assumed  quantities,  we  may  use  any  other  two  quantities  which  may 
be  readily  found  from  the  data  furnished  by  observation.  Thus,  if 
we  compute  r'  and  ur  for  the  date  of  a  third  observation  directly 
from  each  of  the  three  systems  of  elements,  the  differences  between 
the  successive  results  will  furnish  the  numerical  values  of  the  partial 
differential  coefficients  of  rf  and  u'  with  respect  to  J  and  J",  or  with 
respect  to  &  and  i,  as  the  case  may  be.  Then,  computing  the  values 
of  rf  and  uf  from  the  observed  geocentric  spherical  co-ordinates  by 
means  of  the  values  of  Q,  and  i  for  the  system  of  elements  to  be 
corrected,  the  differences  between  the  results  thus  derived  and  thoso 
obtained  directly  from  the  elements  enable  us  to  form  the  equations 

du'  dur       ...  . 

jj"i+3F"r-"' 

• 


328  THEORETICAL   ASTRONOMY. 

or  the  corresponding  expressions  in  the  case  of  the  variation  of  & 
and  i,  by  means  of  which  the  corrections  to  be  applied  to  the  as- 
sumed values  will  be  determined.  In  the  numerical  application  of 
these  equations,  &uf  being  expressed  in  seconds  of  arc,  Ar'  should  also 
be  expressed  in  seconds,  and  the  resulting  values  of  A//  and  A  J"  will 
oe  converted  into  those  expressed  in  parts  of  the  unit  of  space  by 
dividing  them  by  206264.8. 

When  only  three  observed  places  are  to  be  used  for  correcting  an 
approximate  orbit,  from  the  values  of  r,  rf,  r"  and  uf  u',  u"  obtained 
by  means  of  the  formulae  which  have  been  given,  we  may  find  p  and 

a  or the  latter  in  the  case  of  very  eccentric  orbits — from  the  first 

a 

and  second  places,  and  also  from  the  first  and  third  places.  If  these 
results  agree,  the  elements  do  not  require  any  correction;  but  if  a 
difference  is  found  to  exist,  by  computing  the  differences,  in  the  case 
of  each  of  these  two  elements,  for  three  hypotheses  in  regard  to  J 
and  A"  or  in  regard  to  Q>  and  i,  the  equations  may  be  formed  by 
means  of  which  the  corrections  to  be  applied  to  the  assumed  values 
of  the  two  geocentric  distances,  or  to  those  of  &  and  i,  will  be 
obtained. 

110.  The  formulae  which  have  thus  far  been  given  for  the  correc- 
tion of  an  approximate  orbit  by  varying  the  geocentric  distances, 
depend  on  two  of  these  distances  when  no  assumption  is  made  in 
regard  to  the  form  of  the  orbit,  and  these  formulae  apply  with  equal 
facility  whether  three  or  more  than  three  observed  places  are  used. 
But  when  a  series  of  places  can  be  made  available,  the  problem  may 
be  successfully  treated  in  a  manner  such  that  it  will  only  be  necessary 
to  vary  one  geocentric  distance.  Thus,  let  x,  y,  z  be  the  rectangular 
heliocentric  co-ordinates,  and  r  the  radius-vector  of  the  body  at  the 
time  t,  and  let  X,  F,  Z  be  the  geocentric  co-ordinates  of  the  sun  at 
the  same  instant.  Let  the  geocentric  co-ordinates  of  the  body  be 
designated  by  x09  yQ,  ZQ,  and  let  the  plane  of  the  equator  be  taken  as 
the  fundamental  plane,  the  positive  axis  of  x  being  directed  to  the 
vernal  equinox.  Further,  let  p  denote  the  projection  of  the  radius- 
vector  of  the  body  on  the  plane  of  the  equator,  or  the  curtate  dis- 
tance with  respect  to  the  equator;  then  we-shall  have 

x0  =  p  cos  a,  y0  =  p  sin  a,  z0  —  p  tan  d.  (32) 

If  we  represent  the  right  ascension  of  the  sun  by  A,  and  its  declina- 
tion by  _D,  we  also  have 


VARIATION   OF   ONE   GEOCENTRIC   DISTANCE.  329 

Z=RsmD.    (33) 


The  fundamental  equations  for  the  undisturbed  motion  of  the  planet 
or  comet,  neglecting  its  mass  in  comparison  with  that  of  the  sun,  are 


but  since 


and,  neglecting  also  the  mass  of  the  earth, 


these  become 


Substituting  for  a?0,  y0,  and  2;0  their  values  in  terms  of  a  and  d,  and 
putting 


we  get 

§  +  ^coSa  +  ,  =  0, 

+        sina  +  ,  =  0,  '  (36) 


Differentiating  the  equations  (32)  with  respect  to  t,  we  find 
dxn  dp          .      da 

-d?=c 

f 


31)0  THEORETICAL   ASTRONOMY. 

Differentiating  again  with  respect  to  t,  and  substituting  in  the  equa 
tions  (36)  the  values  thus  found,  the  results  are 


If  we  multiply  the  first  of  these  equations  by  sin  a,  and  the  second 
by  —  cos  a,  and  add  the  products,  we  obtain 

d'2a 
j  gBin.-?COB.-/»- 

dt       2  rff 

57 
Now,  from  (35)  we  get 

£  sin  a  —  ^  cos  a  =  &2  1  -=  --  3  I  -K  cos  D  sin  (a  —  A), 
and  the  preceding  equation  becomes 


The  value  of  -vr  thus  found  is  independent  of  the  differential  co- 

dp 

efficients  of  d  with  respect  to  t.     To  find  another  value  of  -J-,  using 

dt 

all  three  of  equations  (38),  we  multiply  the  first  of  these  equations 
by  sin  A  tan  8,  the  second  by  —  cos  A  tan  £,  and  the  third  by 
—  sin  (a  —  A).  Then,  adding  the  products,  since  £  sin  A  =  TJ  cos  A, 
the  result  is 


from  which  we  get 

%  -  cot  (a  -  A)  %  +  «*•*(  2  £'  +  cot  »%  )  +  i  cot* 

a/>  _  ,    <ft«  _  2  _  dt      _  \     dt2    '  _  ^  /   '   p 

= 


VARIATION   OF   ONE   GEOCENTRIC   DISTANCE.  331 

When  the  ecliptic  is  taken  as  the  fundamental  plane,  the  last  term 
of  the  numerator  of  the  second  member  of  this  equation  vanishes, 
and  the  equation  may  be  written 

*=*  (41) 

the  coefficient  C  being  independent  of  p. 

111.  When  the  value  of  p  is  given,  that  of  -=2  will  be  determined 

in  terms  of  the  data  furnished  directly  by  observation  and  of  the 
differential  coefficients  of  a  and  d  with  respect  to  t  from  equation 
(39),  or  from  (40),  the  latter  being  preferred  when  the  motion  of  the 

body  in  right  ascension  is  very  slow.     The  value  of  -77  having  been 

ut 

found,  we  may  compute  the  velocities  of  the  body  in  directions 
parallel  to  the  co-ordinate  axes.  Thus,  since 

x0  =  x  +  X,  y0  =  y  +  Y,  z0  =  z  +  Z, 

the  equations  (37)  give 

dx  dp  .      do,      dX 

dt=co*adi-flSin*dt-W 

dy  dp   ,  da      dY 


_  _  _.__,  (42) 

dz  dp  df      dZ 

_  =  tan^  +  ,seo.3___I 

by  means  of  which  -=-t  -~,  and  -7-  may  be  determined. 

7  -T7*         7  TT-  7  f7 

To  find  the  values  of  -77  >  —  rr>  and  -77,  the  equations 

at     at  at 

X=  Rcos  Q, 
Y=  jRsin  O  cose, 
Z  =  R  sin  O  sin  e, 
give,  by  differentiation, 

dX  dR  ^dO 

^^eosQ..  _jRsm0_, 

^=  sin  Q  cose^  +  R  cos  0  cose^®  (43) 

at  at  at 

dZ  dR   .    n  .dQ 

-,.  =  sm  O  sm  e    ,,  +  R  cos  O  sin  e  -=-. 
at  at  at 


332  THEORETICAL   ASTRONOMY. 

Now,  according  to  equation  (52)1?  we  have 


dQ  _  £T(l-e02)(l  +  m0) 

~dTz  ~ir~ 

mQ  denoting  the  mass  of  the  earth,  and  eQ  the  eccentricity  of  its  orbit. 
The  polar  equation  of  t^ie  conic  section  gives 

dr  _  r2e  sin  v    dv 
~dt~        p       "ST 

Let  -T  denote  the  longitude  of  the  sun's  perigee,  and  this  equation 
gives 

.sin(0-r,    (45) 


If  we  neglect  the  square  of  the  eccentricity  of  the  earth's  orbit,  we 
have  simply 


_ 

-7- 


(4<5) 


The  values  of  -^-  and  -7-  having  been  found  by  means  of  these 

dX  dY 
formulae,  the  equations  (43)  give  the  required  results  for  -—  ,  —  -,  and 

dZ 

-j7,  and  hence,  by  means  of  (42),  we  obtain  the  velocities  of  the 

uit 

comet  or  planet  in  directions  parallel  to  the  co-ordinate  axes. 

112.  The  values  of  x,  y,  and  2  may  be  derived  by  means  of  the 

equations 

x  =  A  cos  <5  cos  a  —  X, 
y  —  A  cos  d  sin  a  —  Y, 
z  =  A  sin  d  —  Z, 

and  from  these,  in  connection  with  the  corresponding  velocities,  the 
elements  of  the  orbit  may  be  found.  The  equations  (32),  give  im- 
mediately the  values  of  the  inclination,  the  semi-parameter,  and  the 
right  ascension  of  the  ascending  node  on  the  equator.  Then,  the 
position  of  the  plane  of  the  orbit  being  known,  we  may  compute  r 
and  u  directly  from  the  geocentric  right  ascension  and  declination  by 
means  of  the  equations  (28)  and  (30).  But  if  we  use  the  values  of 
the  heliocentric  co-ordinates  directly,  multiplying  the  first  of  equa- 
tions (93)x  by  cos  &,  and  the  second  by  sin  &,  and  adding  the  pro- 
ducts, we  have 


VAEIATION   OF   ONE   GEOCENTRIC   DISTANCE.  333 

r  sin  u  =  z  cosec  i, 

r  cos  u  =  x  cos  &  +  y  sin  &  > 

from  which  r  and  w  may  be  found,  the  argument  of  the  latitude  u 
being  referred  to  the  plane  of  xy  as  the  fundamental  plane.     The 

equation 

r2  =  x*  -f  if  +  sa 
gives 

dr_x    dx.y    dy  ,   *     dz 
<ft  ~~f  "  ^       r  'eft  "^f  <«' 
and,  since 

dr  _  r*e  sin  v    cfo  efa  _  kVp 

~dt~     ~^p       "di'  dt~   ~r*~' 

we  shall  have 

V~p    dr 
esmv  =  -~-  .  -T-, 

*     *  (49) 

=  -  —  1, 

from  which  to  find  e  and  v.     Then  the  distance  between  the  peri- 
helion and  the  ascending  node  is  given  by 

to  =  u  —  v. 

The  semi-transverse  axis  is  obtained  from  p  and  e  by  means  of  the 
relation 


Finally,  from  the  value  of  v  the  eccentric  anomaly  and  thence  the 
mean  anomaly  may  be  found,  and  the  latter  may  then  be  referred  to 
any  epoch  by  means  of  the  mean  motion  determined  from  a. 

In  the  case  of  very  eccentric  orbits,  the  perihelion  distance  will  be 
given  by 


and  the  time  of  perihelion  passage  may  be  found  from  v  and  e  by 
means  of  Table  IX.  or  Table  X.,  as  already  illustrated. 

The  equation  (21)t  gives,  if  we  substitute  for  /  its  value  in  terms 
of  p,  denote  by  V  the  linear  velocity  of  the  planet  or  comet,  and  neg- 
lect the  mass, 


Let  if/0  denote  the  angle  which  the  tangent  to  the  orbit  at  the  ex- 
tremity of  the  radius-vector  makes  with  the  prolongation  of  this 
radius-vector,  and  we  shall  have 


334  THEORETICAL   ASTRONOMY. 

dr          dx    ,      dy    ,      dz 


BO  that  the  preceding  equation  gives 

k*p  =  FV  sin2  4<0. 
Hence  we  derive  the  equations 
Vr  sin  4  = 


dx          dy          dz 
from  which  Vr  and  ^0  may  be  found.     Then,  since 

we  shall  have 

P      9Z-2 

1  =  ±1.  _F2,  (51) 

a         r 

by  means  of  which  a  may  be  determined,  and  then  e  may  be  found 
by  means  of  this  and  the  value  of  p. 
The  equations  (49)  and  (50)  give 

F2 
e  sin  (u  —  w)  =  —  r  sin  4>0  cos  40, 


F2 
e  cos  (u  —  w)  =  -  -  r  sin2  4/0  —  1, 


and,  since 


these  are  easily  transformed  into 

2ae  sin  (w  —  a>}  =  (2a  —  r)  sin  240, 

2ae  cos  (it  —  w)  =  —  (2a  —  r)  cos  24>0  —  r. 

If  we  multiply  the  first  of  these  equations  by  —  cos  u  and  the  second 
by  sin  w,  and  add  the  products;  then  multiply  the  first  by  sin  it  and 
the  second  by  cos  it,  and  add,  we  obtain 

2ae  sin  w  =  —  (2a  —  r)  sin  (2^0  +  w)  —  r  sin  it,  /  ^N 

2ae  cos  w  =  —  (2a  —  r)  cos  (2^0  +  u)  —  r  cos  u, 

These  equations  give  the  values  of  a)  and  e. 

113.  We  have  thus  derived  all  the  formulae  necessary  for  finding 
the  elements  of  the  orbit  of  a  heavenly  body  from  one  geocentric 
distance,  provided  that  the  first  and  second  differential  coefficients  of 
a  and  d  with  respect  to  the  time  are  accurately  known.  It  remains, 


VARIATION   OF   ONE   GEOCENTRIC   DISTANCE.  335 

therefore,  to  devise  the  means  by  which  these  differential  coefficients 
may  be  determined  with  accuracy  from  the  data  furnished  by  obser- 
vation. The  approximate  elements  derived  from  three  or  from  a 
small  number  of  observations  will  enable  us  to  correct  the  entire 
series  of  observations  for  parallax  and  aberration,  and  to  form  the 
normal  places  which  shall  represent  the  series  of  observed  places. 
We  may  now  assume  that  the  deviation  of  the  spherical  co-ordinates 
computed  by  means  of  the  approximate  elements  from  those  which 
would  be  obtained  if  the  true  elements  were  used,  may  be  exactly 
represented  by  the  formula 

*0  =  A  +  Bh+  Ch\  (53) 

h  denoting  the  interval  between  the  time  at  which  the  deviation  is 
expressed  by  A  and  the  time  for  which  this  difference  is  A#.  The 
differences  between  the  normal  places  and  those  computed  with  the 
approximate  elements  to  be  corrected,  will  then  suffice  to  form  equa- 
tions of  condition  by  means  of  which  the  values  of  the  coefficients 
A,  B,  and  C  may  be  determined.  The  epoch  for  which  h  =  0  may 
be  chosen  arbitrarily,  but  it  will  generally  be  advantageous  to  fix  it 
at  or  near  the  date  of  the  middle  observed  place.  If  three  observed 
places  are  given,  the  difference  between  the  observed  and  the  com- 
puted value  of  each  right  ascension  will  give  an  equation  of  condition, 
according  to  (53),  and  the  three  equations  thus  formed  will  furnish 
the  numerical  values  of  A,  J3,  and  C.  These  having  been  deter- 
mined, the  equation  (53)  will  give  the  correction  to  be  applied  to  the 
computed  right  ascension  for  any  date  within  the  limits  of  the 
extreme  observations  of  the  series.  When  more  than  three  normal 
places  are  determined,  the  resulting  equations  of  condition  may  be 
reduced  by  the  method  of  least  squares  to  three  final  equations,  from 
which,  by  elimination,  the  most  probable  values  of  A,  j5,  and  C  will 
bo  derived.  In  like  manner,  the  corrections  to  be  applied  to  the 
computed  latitudes  may  be  determined.  These  corrections  being 
applied,  the  ephemeris  thus  obtained  may  be  assumed  to  represent 
the  apparent  path  of  the  body  with  great  precision,  and  may  be  em- 
ployed as  an  auxiliary  in  determining  the  values  of  the  differential 
coefficients  of  a  and  d  with  respect  to  t. 

Let  f(a)  denote  the  right  ascension  of  the  body  at  the  middle 
epoch  or  that  for  which  h  =  Q,  and  let /(a  dz  no))  denote  the  value  of 
a  for  any  other  date  separated  by  the  interval  na)9  in  which  at  is  the 
interval  between  the  successive  dates  of  the  ephemeris.  Then,  if  we 
put  n  successively  equal  to  1,  2,  3,  &c.,  we  shall  have 


336  THEORETICAL   ASTRONOMY. 

Function.          I.  Diff.  II.  Diff.  III.  Diff.         IV.  Diff.          V.  Diff. 

/(a  —  3a>)  ff 


/(a  -f  3a>) 

The  series  of  functions  and  differences  may  be  extended  in  the  same 
manner  in  either  direction.  If  we  expand  /(a  +  nco)  into  a  series. 
the  result  is 

/(a-ftt«0^:a-|-~nw  +  2^w2ty2  +  £^wVJ4-  A  ^•^*cy*  +  &c-» 

or,  putting  for  brevity  A  =  -j-a))  B  =  \  -^  -  a)2,  &c., 

/(a  +  nco)  =  a  -f  An  -f  5^2  +  Oi3  -f  Dn4  +  &c. 
If  we  now  put  n  successively  equal  to  —  4,  —  3,  —  2,  —  1,  —  0,  +1, 


&c.,  we  obtain  the  values  of  /(a  —  4co),f(a  —  3w),  ......  /(a  -f  4a>) 

in  terms  of  A,  B,  C,  &c.  Then,  taking  the  successive  orders  of 
differences  and  symbolizing  them  as  indicated  above,  we  obtain  a 
series  of  equations  by  means  of  which  A,  B,  C,  &c.  will  be  deter- 
mined in  terms  of  the  successive  orders  of  differences.  Finally,  re- 
placing Ay  B,  C,  &G.  by  the  quantities  which  they  represent,  and 
putting 

ir  (<*-*«)  +  if  '(<*+j«o  =/'(«), 

if'"  («  -  *-)  +•  if  '"  («  +  i»)  =/"'  (a),  &c.( 
we  obtain 

=  A-  (/'(a)  -  I/'"  (a)  +  3>5/'(oO  -  Ti^f-(a)  +  &c.), 


=       (/"(«)  -  */"(«)  +  *?5/""  («)  -  *<>.), 


=       (/"(«)  -1/-W 


VARIATION    OF   ONE   GEOCENTRIC   DISTANCE.  337 

by  means  of  which  the  successive  differential  coefficients  of  a  with 
respect  to  t  may  be  determined.  The  derivation  of  these  coefficients 
in  the  case  of  d  is  entirely  analogous  to  the  process  here  indicated  for 
a.  Since  the  successive  differences  will  be  expressed  in  seconds  of 
arc,  the  resulting  values  of  the  differential  coefficients  of  a  and  d  with 
respect  to  t  will  also  be  expressed  in  seconds,  and  must  be  divided  by 
^06264.8  in  order  to  express  them  abstractly. 

We  may  adopt  directly  the  values  of  -JT>  -^p  -yr.  and  -^  determined 

by  means  of  the  corrected  ephemeris,  or,  if  the  observed  places  do 
not  include  a  very  long  interval,  we  may  determine  only  the  values 

of  -jp,  -j£,  &G.  by  means  of  the  ephemeris,  and  then  find  -=-  and  -^- 

directly  from  the  normal  places  or  observations.  Thus,  let  a,  a/,  a" 
be  three  observed  right  ascensions  corresponding  to  the  times  t,  t',  t", 
and  we  shall  have 


which  give 


These  equations,  being  solved  numerically,  will  give  the  values  of  -7- 

d*a 

and  —  ,  and  we  may  thus  by  triple  combinations  of  the  observed 

places,  using  always  the  same  middle  place,  form  equations  of  con- 
dition for  the  determination  of  the  most  probable  values  of  these 
differential  coefficients  by  the  solution  of  the  equations  according  to 
the  method  of  least  squares. 

7  »  /72/^ 

In  a  similar  manner  the  values  of  -^  and  -jr  may  be  derived. 

114.  In  applying  these  formulae  to  the  calculation  of  an  orbit, 
after  the  normal  places  have  been  derived,  an  ephemeris  should  be 
computed  at  intervals  of  four  or  eight  days,  arranging  it  so  that  one 
of  the  dates  shall  correspond  to  that  of  the  middle  observation  or 
normal  place.  This  ephemeris  should  be  computed  with  the  utmost 


338  THEORETICAL   ASTRONOMY. 

care,  since  it  is  to  be  employed  as  an  auxiliary  in  determining  quan- 
tities on  which  depends  the  accuracy  of  the  final  results.  The  com- 
parison of  the  ephemeris  with  the  observed  places  will  furnish,  by 
means  of  equations  of  the  form 

A  +  Eh  +  Ch2  =  Aa', 
A  _f-  B'h  +  C'K  =  A*, 

k  being  the  interval  between  the  middle  date  tf  and  that  of  the  place 
used,  the  values  of  A,  J5,  (7,  A',  &c.;  and  the  corrections  to  be 
applied  to  the  ephemeris  will  be  determined  by 

A  -f  Bna>  -f  Oi2"8  =  Aa, 
A' 


The  unit  of  h  may  be  ten  days,  or  any  other  convenient  interval, 
observing,  however,  that  no)  in  the  last  equations  must  be  expressed 
in  parts  of  the  same  unit.  With  the  ephemeris  thus  corrected,  we 

«da  d?a  dd       _  d?8 
compute  the  values  of  -j-,  -^-,  ^7,  and  j-  as  already  explained,     inese 

differential  coefficients  should  be  determined  with  great  care,  since  it 
is  on  their  accuracy  that  the  subsequent  calculation  principally  de- 

.  .      dX  dY       ,  dZ  . 

pends.     We  compute,  also,  the  velocities  -jrt  -rr,  and  —rr  by  means 
7x-v  7  -p  dt     at  at 

of  the  formulaB  (43),  —^-  and  —  being  computed  from  (46).     The 

quantities  thus  far  derived  remain  unchanged  in  the  two  hypotheses 
with  regard  to  J. 

Then  we  assume  an  approximate  value  of  J,  and  compute 

p  —  J  cos  d  ; 

and  by  means  of  the  equation  (40)  or  (39)  we  compute  the  value  ol 
-—•  It  will  be  observed  that  if  we  put  the  equation  (40)  in  the  form 

dp      P          C 
_  =  _,  +  _ 

p 

the  coefficient  -^  remains  the  same  in  the  two  hypotheses.     The  three 

equations  (38)  may  be  so  combined  that  the  resulting  value  of  — 

_J2  ^*^ 

will  not  contain  -T~>     This  transformation  is  easily  effected,  and  may 

(J  tL 

be  advantageous  in  special  cases  for  which  the  value  of  -^  is  very 

uncertain. 

The  heliocentric  spherical  co-ordinates  will  be  obtained  from  the 


RELATION    BETWEEN   TWO   PLACES   IN   THE   OEBIT.  339 

assumed  value  of  J  by  means  of  the  equations  (106)31  and  the  rec- 
tangular co-ordinates  from 

x  =  r  cos  b  cos  I, 
y  =  r  cos  b  sin  I, 
z  ==r  sin  b. 

The  velocities  -57,  ~,  and  -r-  will  be  given  by  (42),  and  from  these 

and  the  co-ordinates  x,  y,  z  the  elements  of  the  orbit  will  be  com- 
puted by  means  of  the  equations  (32)1?  (47),  (49),  &c.  With  the 
elements  thus  derived  we  compute  the  geocentric  places  for  the  dates 
of  the  normals,  and  find  the  differences  between  computation  and 
observation.  Then  a  second  system  of  elements  is  computed  from 
J  4-  $d,  and  compared  with  the  observed  places.  Let  the  difference 
between  computation  and  observation  for  either  of  the  two  spherical 
co-ordinates  be  denoted  by  n  for  the  first  system  of  elements,  and  by 
nf  for  the  second  system.  The  final  correction  to  be  applied  to  J,  in 
order  that  the  observed  place  may  be  exactly  represented,  will  be 
determined  by 

££(»'_„) +  n-0.  (56) 

Each  observed  right  ascension  and  each  observed  declination  will 
thus  furnish  an  equation  of  condition  for  the  determination  of  A  J, 
observing  that  the  residuals  in  right  ascension  should  in  each  case  be 
multiplied  by  cos  d.  Finally,  the  elements  which  correspond  to  the 
geocentric  distance  J  -f-  A  A  will  be  determined  either  directly  or  by 
interpolation,  and  these  must  represent  the  entire  series  of  observed 
places. 

115.  The  equations  (52)s  enable  us  to  find  two  radii-vectores  when 
the  ratio  of  the  corresponding  curtate  distances  is  known,  provided 
that  an  additional  equation  involving  r,  r",  K,  and  known  quantities 
is  given.  For  the  special  case  of  parabolic  motion,  this  additional 
equation  involves  only  the  interval  of  time,  the  two  radii-vectores, 
and  the  chord  joining  their  extremities.  The  corresponding  equation 
for  the  general  conic  section  involves  also  the  semi-transverse  axis 
of  the  orbit,  and  hence,  if  the  ratio  M  of  the  curtate  distances  is 
known,  this  equation  will,  in  connection  with  the  equations  (52)3, 
enable  us  to  find  the  values  of  r  and  r"  corresponding  to  a  given 
value  of  a.  To  derive  this  expression,  let  us  resume  the  equations 


THEORETICAL    ASTRONOMY. 

4  =  E"  -  E  -  2e  sin  £  (E"  -  E}  cos  J  (£"  + 
a? 

r  -f  r"  =  2a  —  2ae  cos  £  (£"  —  £)  cos  J  (£"  -f-  JS). 
For  the  chord  x  we  have 

x1  =  (r  +  r")2  —  4rr"  cos*  J  (u"  —  w), 
which,  by  means  of  (58)4,  gives 


cos4 

and,  substituting  for  r  +  ^r/  its  value  given  by  the  last  of  equations 
(57),  we  get 


x2  =  4a2  sin2  £  (£"  —  JE)  (1  —  e2  cos2  &E'  +  E)).  (58) 

J^et  us  now  introduce  an  auxiliary  angle  ^,  such  that 

cos  h  =  e  cos  £  CE"  H-  J5;), 
the  condition  being  imposed  that  h  shall  be  less  than  180°,  and  put 


then  the  equations  (57)  and  (58)  become 

—  =  2g  —  2  sin  g  cos  h, 

r  +  r"  =  2a  (1  —  cos  g 
x  =  2a  sin  ^  sin  h. 


"  =  2a  (1  —  cos  g  cos  A), 


Further,  let  us  put 

h  —  ff  =  $, 

and  the  last  two  of  equations  (59)  give 


Introducing  £  and  e  into  the  first  of  equations  (59),  it  becomes 


—s  =  0  —  sin  e)  —  (<5  —  sin  5).  (61) 

a? 

The  formulae  (60)  enable  us  to  determine  e  and  d  from  r  -f  r",  x, 
and  a,  and  then  the  time  r'  =  k  (tff  —  t)  may  be  determined  from 
(61).  Since,  according  to  (58)4, 

Vrr"  cos  J  (u"  —  u)  =  a  (cos  ^  —  cos  7i)  =  2  sin  ^e  Bin  £<?, 


EELATION   BETWEEN   TWO   PLACES   IN   THE   ORBIT.  341 

and  since  sin  %s  is  necessarily  positive,  it  appears  that  when  un  —  it 
exceeds  180°,  the  value  of  sin  |<5  must  be  negative,  and  when 
u"  —  16  =  180°,  we  have  S  =  Q;  and  thus  the  quadrant  in  which 
d  must  be  taken  is  determined.  It  will  be  observed  that  the  value 
of  Je,  as  given  by  the  first  of  equations  (60),  may  be  either  in  the 
first  or  the  second  quadrant;  but,  in  the  actual  application  of  the 
formulae,  the  ambiguity  is  easily  removed  by  means  of  the  known 
circumstances  in  regard  to  the  motion  of  the  body  during  the  in- 
terval t"  —  t. 

In  the  application  of  the  equations  (52)3,  by  means  of  an  approxi 
mate  value  of  x  we  compute  d,  and  thence  r  and  r".    Then  we  com- 
pute £  and  S  corresponding  to  the  given  value  of  a,  and  from  (61) 
we  derive  the  value  of 


If  this  agrees  with  the  observed  interval  t"  —  £,  the  assumed  value 
of  x  is  correct;  but  if  a  difference  exists,  by  varying  x  we  may 
readily  find,  by  a  few  trials,  the  value  which  will  exactly  satisfy  the 
equations.  The  formulae  (70)3  will  then  enable  us  to  determine  the 
curtate  distances  p  and  p",  and  from  these  and  the  observed  spherical 
co-ordinates  the  elements  of  the  orbit  may  be  found. 

As  soon  as  the  values  of  u  and  u"  have  been  computed,  since 
£  —  d  =  E"  —  Ej  we  have,  according  to  equation  (85)4, 

sin  i  (u"  —  M)  ./—  77 
cos  <[>  =  —  r^-T  -  ~  V  rr", 
—  8) 


which  may  be  used  to  determine  <p  when  the  orbit  is  very  eccentric. 
To  find  p  and  q,  we  have 

p  =  a  cos2  <p,  q  =  2a  sin2  (45°  —  £p)  ; 

and  the  value  of  to  may  be  found  by  means  of  the  equations  (87)4  or 
(88), 

116.  The  process  here  indicated  will  be  applied  chiefly  in  the  de- 
termination of  the  orbits  of  comets,  and  generally  for  cases  in  which 
a  is  large.  In  such  cases  the  angles  £  and  d  will  be  small,  so  that 
the  slightest  errors  will  have  considerable  influence  in  vitiating  the 
value  of  t"  —  t  as  determined  by  equation  (61);  but  if  we  transform 
this  equation  so  as  to  eliminate  the  divisor  a%  in  the  first  member,  the 
uncertainty  of  the  solution  may  be  overcome.  The  difference  £  —sine 


342  THEORETICAL   ASTRONOMY. 

may  be  expressed  by  a  series  which  converges  rapidly  when  e  is  small. 
Thus,  let  us  put 

e  —  sin  s  =  y  sin8  ^e,  x  =  sin2  \et 

and  we  have 

-j~  =  2  cosec  %e  —  |y  cot  Je, 

i   .  de 

— —  =  4  cosec  ^s. 

Therefore 

dy  _  8  —  6y  cos  |e  _  4  •—  3y  (1  —  2aQ 
eta  sin2  Je  2#  (1  —  a;) 

If  we  suppose  y  to  be  expanded  into  a  series  of  the  form 

y  =  a  +  fa  -r  r**  +  ^  +  &C., 

we  get,  by  differentiation, 


ATP 

~ j         — —  f*    — r~    s-ij  »v   — j~~    v'-'**/      — r™   CXv» j 

and  substituting  for  -7—  the  value  already  obtained,  the  result  is 

2#B  -f  (4r  —  2/9)  #2  -f  (65  —  4r)  z3  +  Ac.  =  4  —  3a  -f  (6»  —  3/9)  x 

+  (6/9  —  3r)  a2  +  (6r  —  35)  x9  +  Ac. 
Therefore  we  have 

4  —  3<*  =  0,  6a  — 3/9=2/9, 

from  which  we  get 

4^_6  _4.6.8         J  =  4.6.8;iO  &g 

Hence  we  obtain 


.  4*«lf-11Jt*41        1*41         1*81         |jf  1^/3  O^ 

f  —  sm s  =  I  sin3 ^e  I  l-{-|sin3|£-|-p — =sm*^e-| — _   _  ^  snr^£-f-<Kc.  j,  (^b^; 

and,  in  like  manner, 

8 — sin  d  =  |  sin3  %dl  l-f-gsin2j5-[-— ^=sinAj^-|"  g*  7'  Q  sin6  |5-{-Ac. ),  (63) 

which,  for  brevity,  may  be  written 

e  —  sin  e  =  |  Q  sin8  £e, 


RELATION   BETWEEN   TWO   PLACES   IN   THE   ORBIT.  343 

Combining  these  expressions  with  (61),  and  substituting  for  sin^e  and 
Bin  |<5  their  values  given  by  the  equations  (60),  -there  results 

6r'  =  Q  (r  +  r"  +  x)t  +  Q'  (r  +  r"  -  x)f  ,  (65) 

the  upper  sign  being  used  when  the  heliocentric  motion  of  the  body 
is  less  than  180°,  and  the  lower  sign  when  it  is  greater  than  180°. 
The  coefficients  Q  and  Qf  represent,  respectively,  the  series  of  terms 
enclosed  in  the  parentheses  in  the  second  members  of  the  equations 
(62)  and  (63),  and  it  is  evident  that  their  values  may  be  tabulated 
with  the  argument  e  or  d}  as  the  case  may  be.  It  will  be  observed, 
however,  that  the  first  two  terms  of  the  value  of  Q  are  identical  with 
the  first  two  terms  of  the  expansion  of  (cos  Je)"~  *  into  a  series  jf 
ascending  powers  of  sin  Je,  while  the  difference  is  very  small  between 
the  coefficients  of  the  third  terms.  Thus,  we  have 

(cos  Je)~  V2  =  (1  —  sin2  Je)-*  =  1  +  f  sin2  je  +  |^  sin4  Je 

O  .  1U 

.  6  .  11  .  16  . 


and  if  we  put 

g=       B'»,  (66) 

(cos  J«)  * 

we  shall  have 

£o  =  l  +  T?  5  sin*  i*  +  2V-f5  sin6  Je  -f  Ac.  (67) 

In  a  similar  manner,  if  we  put 

«=(^  (68) 

we  find 

-Bo'  =  1  +  if  5  sin4  i*  +  JJ&  sin6  15  +  &c.  (69) 

Table  XV.  gives  the  values  of  BQ  or  BQr  corresponding  to  £  or  d  from- 
0°  to  60°. 

For  the  case  of  parabolic  motion  we  have 


and  the  equation  (65)  becomes  identical  with  (56)8. 

In  the  application  of  these  formulae,  we  first  compute  e  and  d  by 
means  of  the  equations  (60),  and  then,  having  found  BQ  and  B9'  by 
means  of  Table  XV.,  we  compute  the  values  of  Q  and  Q'  from  (66) 
and  (68).  Finally,  the  time  r'  =  k  (t"  —  t)  will  be  obtained  from  (65), 
and  the  difference  between  this  result  and  the  observed  interval  will 


344  THEORETICAL   ASTRONOMY. 

indicate  whether  the  assumed  value  of  K  must  be  increased  or  di- 
minished.    A  few  trials  will  give  the  correct  result. 

117.  Since  the  interval  of  time  I"  —  t  cannot  be  determined  with 
sufficient  accuracy  from  (65)  when  the  chord  x  is  very  small,  it 
becomes  necessary  to  effect  a  further  transformation  of  this  equation. 
Thus,  let  us  put 

Q  —  q  =  6P,  x  =  sin2  Je,  a?  =  sin1  JJ, 

and  we  shall  have 


Now,  when  x  is  very  small,  we  may  put 

cos|e  =  cos;J<J, 
and  hence 

,        .  .  ,          •  ,  i*      sin2  ^£  —  si 
x  —  x'  =  sin2  4e  —  sin*i3  =  -  f  -  =-= 

4  cos2  | 

which,  by  means  of  equations  (60),  becomes 

r  * 

X  -  X  =  5  --  r-:—  • 

oacos  3* 

Therefore  we  have,  when  x  is  very  small, 

P=40aL-i«  (1  +  V  *™'  i£  +  1?  8in'  is 
If  we  put 


t 

the  equation  (65)  becomes,  using  only  the  upper  sign, 

(r  +  /'  +  X)f  -  (r  +  r"  -  x)f  =  6r0',  (72) 

which  is  of  the  same  form  as  (56)3.     Hence,  according  to  the  equa- 
tions (63)s  and  (66)3,  we  shall  have 

*=<"  (73) 


toe  value  of  //  being  found  from  Table  XI.  with  the  argument 


EELATION    BETWEEN   TWO   PLACES   IN   THE   OKBIT.  345 

It  remains,  therefore,  simply  to  find  a  convenient  expression  for  r/, 
and  the  determination  of  x  is  effected  by  a  process  precisely  the  same 
as  in  the  special  case  of  parabolic  motion. 
Let  us  now  put 

N 


and  we  shall  have 

cos2je  /1    ,  2.  8  .          .3.8.10.         .4.8.10.12.         .  .     V 
N=—-    l  +     ~sm  i£  +  "~  Sm4i£+--sm6ie+&c-  • 


or,  substituting  for  Q  its  value  in  terms  of  sin  Je, 

JV=  1  +  A  sin2  J-e  +  ^  sin*  Je  +  ^%  sin«  J.  +  &c.         (75) 
Therefore^  if  we  put 


the  expression  for  r0'  becomes 


(77) 


Table  XV.  gives  the  value  of  log  N  corresponding  to  values  of  e 
from  e  =  0  to  e  =  60°. 

If  the  chord  K  is  given,  and  the  interval  of  time  t"  —  t  is  required, 
we  compute  Ar/  by  means  of  (76),  and,  having  found  r/  from 


as  in  the  case  of  parabolic  motion,  we  have 


It  should  be  observed  that  although  equation  (76)  is  derived  for  the 
case  of  a  small  value  of  x,  yet  it  is  applicable  whenever  the  differ- 
ence £  —  d  is  very  small,  whatever  may  be  the  value  of  K.  For 
orbits  which  differ  but  little  from  the  parabolic  form,  it  will  in  all 
cases  be  sufficient  to  use  this  expression  for  Ar/;  and  for  cases  in 
which  the  difference  between  e  and  3  is  such  that  the  assumption  of 
cos  \s  =  cos  {3,  x  +  xr  =  2x,  &c.,  made  in  deriving  equation  (70),  does 


346  THEORETICAL   ASTRONOMY. 

not  afford  the  required  accuracy,  we  may  compute  both  Q  and  Q' 
directly,  and  then  we  have 

•  +  r"-x)f.  (78) 

The  values  of  the  factor  |  1  1 -~-  I  may  be  tabulated  directly  with 

r-f-r"  x 

— -r —  as  the  vertical  argument  and  -j-  as  the  horizontal  argument; 

but  for  the  few  cases  in  which  the  value  of  N  given  by  the  equation 
(75)  is  not  sufficiently  accurate,  it  will  be  easy  to  compute  Q  and  Q' 
by  means  of  the  formulae  (66)  and  (68),  and  then  find  Ar0'  from  (78). 
Further,  when  there  is  any  doubt  as  to  the  accuracy  of  the  result 
given  by  (76),  for  the  final  trial  in  finding  x  from  r  -f  r"  and  r0  by 
means  of  the  equations  (73)  and  (74),  it  will  be  advisable  to  compute 
Ar0'  from  (78). 

It  appears,  therefore,  that  for  nearly  all  the  cases  which  actually 
occur  the  determination  of  the  value  of  x,  corresponding  to  given 

values  of  a  and  Jf  =  — ,  is  reduced  by  means  of  the  equation  (72)  to 

the  method  which  is  adopted  in  the  case  of  parabolic  orbits. 

The  calculation  of  the  numerical  values  of  r  -}-  r"-{-  x  and  r  +  rf/ —  K 
will  be  most  conveniently  effected  by  the  aid  of  addition  and  sub- 
traction logarithms.  If  the  tables  of  common  logarithms  are  used, 
we  may  first  compute 


and  then  we  have 


r  +  r"  +  x  =  2  (r  +  r")  sin2  (45°  +  J 
r  +  r"  —  x  =  2  (r  +  r")  cos2  (45°  + 


118.  In  the  case  of  hyperbolic  motion,  the  semi-transverse  axis  is 
negative,  and  the  values  of  sin  }e  and  sin  \8  given  by  the  equations 
(60)  become  imaginary,  so  that  it  is  no  longer  possible  to  compute 
the  interval  of  time  from  r  +  r"  and  x  by  means  of  the  auxiliary 
angles  e  and  d.  Let  us,  therefore,  put 

sin2  ^e  =  —  m2,  sin2  $3  =  —  n*  ; 

then,  when  a  is  negative,  m  and  n  will  be  real.     Now  we  have 

£e  =  sin""1  V—  m\  j,d  =  sin"1  V^n\ 

and 

-|e  V^I  =  loge  (cos  \*  4-  T  /^1  sin  £«). 


RE].ATIOIs    BETWEEN    TWO   PLACES   IN   THE   ORBIT.  347 

Hence  we  derive 


e  =  2sin      V  —  m2  =  -7=  loge  (1/1  +  m2  +  w), 

d  =  2  sin  -1  l/^T2  =  —  =  loge  (l/l  -j-  n2  +  n). 
X  —  1 

Substituting  these  values  in  the  equation  (61),  and  writing  —  a  in- 
stead of  a,  since 

sin  e  =  2m  I/—  1  •  1/1  -f-  w3, 
we  shall  have 

~  =  2m  l/l  -f-  m2  —  2  loge  (1/1  +  m2  +  m) 


=P  (2n  11  +  n*  —  2  loge 

the  upper  sign  being  used  when  the  heliocentric  motion  is  less  than 
180°,  and  the  lower  sign  when  it  is  greater  than  180°.  Therefore, 
if  we  compute  m  and  n  from 


regarding  the  hyperbolic  semi-transverse  axis  a  as  positive,  the  for- 
mula (79)  will  determine  the  interval  of  time  r'  =  k  (tff  —  t). 

The  first  two  terms  of  the  second  member  of  equation  (79)  may 
be  expressed  in  a  series  of  ascending  powers  of  m,  and  the  last  two 
terms  in  a  series  of  ascending  powers  of  n.  Thus,  if  we  put 

loge  (/I  -f  m2  -f  m)  ==  am  -f-  /?m2  +  r™?  +  ^  +  &c., 
we  get,  by  differentiation, 
1 


=  =  a  +  2/?m  +  3rm2  -f  45m3  +  5em*  +  &c. ; 
m2 

and  since 

1  1-3  1-3-5  „ 


we  have 


Hence  we  obtain 


2  log,  (l/iT^2"-!-  m)  =  2m  —  Jm8  +  i  •  |wi6  -  |         m7  +  &c. 


348  THEORETICAL   ASTRONOMY. 

We  have,  also, 


2m  l/l  -f  m2  =  2m  -f  ms  —  |m5  +  j  -^m7  —  &c. 
Therefore, 


and  similarly 

2?i  1/1  -f-  ^  —  2  loge  (l/l  -f  n*  +  n)  = 


2m  1/1  +  m2  —  2  loge  (l/l  +  m2  -f  m)  = 


(82) 


Substituting  these  values  in  the  equation  (79),  and  denoting  the 
series  of  terms  enclosed  in  the  parentheses  by  Q  and  §',  respectively, 
we  get 

6r'  =  g  (r  +  r"  +  x)l  +  §'  (r  -f  r"  -  x)f  (83) 

which  is  identical  with  equation  (65).  If  we  replace  m2  by  —  sin2|e 
and  n2  by  —  sin2^  in  the  expressions  for  Q  and  Q',  as  given  by  (81) 
and  (82),  we  shall  have  the  expressions  for  these  quantities  in  terms 
of  sin  |e  and  sin  |<J,  respectively,  instead  of  sin  %s  and  sin  J<5  as  given 
by  the  equations  (62)  and  (63),  namely, 

Q  =1  +  f  -isin'1,  +  |  ljsin'-ie  +  |il4.Siii«^  +  &c.( 


For  the  case  of  an  elliptic  orbit  it  is  most  convenient  to  use  the 
equations  (66)  and  (68)  in  finding  Q  and  Qf  ;  but,  since  the  cases  of 
hyperbolic  motion  are  rare,  while  for  those  which  do  occur  the  eccen- 
tricity is  very  little  greater  than  that  of  the  parabola,  it  will  be  suf- 
ficient to  tabulate  Q  directly  with  the  argument  m.  The  same  table, 
using  n  as  the  argument,  will  give  the  value  of  Qf.  Table  XVI. 
gives  the  values  of  Q  corresponding  to  values  of  m  from  m  =  0  to 
m  =  0.2. 

When  the  values  of  r  -f  r",  rf,  and  a  are  given,  and  the  chord  x 
is  required,  we  may  compute  Ar/  from  (78),  T0f  from  (77),  and  finally 
#  from  (73). 

It  may  be  remarked,  also,  that  the  formulae  for  the  relation  between 
T',  r-\-r",  K,  and  a  suffice  to  find  by  trial  the  value  of  a  when  r-fr" 
and  x  are  given.  Hence,  in  the  computation  of  an  orbit  from  assumed 


RELATION    BETWEEN    TWO    PLACES    IN    THE   ORBIT.  049 

values  of  J  and  J/r,  the  value  of  X  may  be  computed  from  r,  rfty  and 
u"  -  u,  and  then  a  may  be  found  in  the  manner  here  indicated. 

If  we  substitute  in  the  equations  (84)  the  values  of  sin  ^s  and  sin  \d 
in  terms  of  r  4-  r",  x,  and  a,  and  then  substitute  the  resulting  values 
of  Q  and  Qf  in  the  equation  (65),  we  obtain 


+  sfe  ^  ((r  +  f"  +  x)*  qF  fr  +  *"  -  x)*)  +  &c.f 

the  lower  sign  being  used  when  un  —  u  exceeds  180°.  When  the 
eccentricity  is  very  nearly  equal  to  unity,  this  series  converges  with 
great  rapidity.  In  the  case  of  hyperbolic  motion,  the  sign  of  a  must 
be  changed. 

119.  The  formulae  thus  derived  for  the  determination  of  the  chord  x 
for  the  cases  of  elliptic  and  hyperbolic  orbits,  enable  us  to  correct  an 
approximate  orbit  by  varying  the  semi-transverse  axis  a  and  the 
ratio  M  of  two  curtate  distances.  But  since  the  formulae  will  gene- 
rally be  applied  for  the  correction  of  approximate  parabolic  elements, 

or  those  which  are  nearly  parabolic,  it  will  be  expedient  to  use  -  and 

M  as  the  quantities  to  be  determined. 

In  the  first  place,  we  compute  a  system  of  elements  from  M  and 

/=-;  and,  for  the  determination  of  the  auxiliary  quantities  pre- 

CL 

liminary  to  the  calculation  of  the  values  of  r,  r",  and  x,  the  equa- 
tions (41)s,  (50)s,  and  (51)3  will  be  employed  when  the  ecliptic  is  the 
fundamental  plane.  But  when  the  equator  is  taken  as  the  funda- 
mental plane,  we  must  first  compute  #,  K,  and  G  by  means  of  the 
equations  (96)3.  Then,  by  a  process  entirely  analogous  to  that  by 
which  the  equations  (47)3  and  (50)3  were  derived,  we  obtain 

h  cos  C  cos  (H—  a")  =  M—  cos  (a"  —  a), 

h  cos  C  sin  (H—  a")  =  sin  (a"  —  a),  (86) 

h  sin  C  =M  tan  8"  —  tan  d, 

from  which  to  find  H,  f,  and  h;  and  also 

cos  <p  =  cos  C  cos  Kcos(G  —  H)  +  sin  C  sin  K,  (87) 

from  which  to  find  <f>.  In  this  case,  £  and  H  will  be  referred  to  the 
equator  as  the  fundamental  plane.  The  angles  ^  and  i//'  will  be 
obtained  from  the  equations  (102)3,  or  from  equations  of  the  form 


350  THEORETICAL   ASTRONOMY. 

of  (26),  and  finally  the  auxiliary  quantities  A,  B,  B",  &c.  will  be 
obtained  from  (51)3,  writing  d  and  d/f  in  place  of  ft  and  ft",  respect- 
ively. 

As  soon  as  these  auxiliary  quantities  have  been  determined,  by 
means  of  (52)3  the  value  of  %  must  be  found  which  will  exactly 
satisfy  equation  (65).  To  effect  this,  we  first  compute  e  from 


sin  ^e  = 
and,  if  it  be  required,  we  also  find  3  from 


using  approximate  values  of  r  +  r"  and  x.  Then  we  find  Q  from 
(66),  and  Ar0'  from  (76)  or  from  (78),  the  logarithms  of  the  auxiliary 
quantities  BQ  and  N  being  found  by  means  of  Table  XV.  with  the 
argument  e.  The  value  of  r/  having  been  found  from  (77),  the 
equations  (73)  and  (74),  in  connection  with  Table  XI.,  enable  us  to 
obtain  a  closer  approximation  to  the  correct  value  of  x.  With  this 
we  compute  new  values  of  r  and  rff,  and  repeat  the  determination 
of  x.  A  few  trials  will  generally  give  the  correct  result,  and  these 
trials  may  be  facilitated  by  the  use  of  the  formula  (67)3.  It  will  be 
observed,  also,  that  Q  and  Ar0'  are  very  slightly  changed  by  a  small 
change  in  the  values  of  r  -j-  rfl  and  x,  so  that  a  repetition  of  the 
calculation  of  these  quantities  pnly  becomes  necessary  for  the  final 
trial  in  finding  the  value  of  x  which  completely  satisfies  the  equa- 
tions (52)3  and  (65).  When  the  value  of  a  is  such  that  the  values 
of  Q  and  ^V  exceed  the  limits  of  Table  XV.,  the  equation  (61)  may 
be  employed,  and,  in  the  case  of  hyperbolic  motion,  when  Q  and  §' 
exceed  the  limits  of  Table  XVI.,  we  may  employ  the  complete  ex- 
pression for  the  time  r'  in  terms  of  m  and  n  as  given  by  (79). 

The  values  of  r,  r/r,  and  x  having  thus  been  found,  the  equations 


will  determine  the  curtate  distances  p  and  />".  When  the  equator  is 
the  fundamental  plane,  we  have 

p  =  A  cos  9,  p"  =  J"  cos  d". 

From  p,  p",  and  the  corresponding  geocentric  spherical  co-ordinates, 
the  radii-  vectores  and  the  heliocentric  spherical  co-ordinates  I,  I",  6, 
and  b"  will  be  obtained,  and  thence  &,  i,  u,  uff,  and  the  remaining 


VARIATION   OF   THE   SEMI-TRANSVERSE   AXIS.  351 

elements  of  the  orbit,  as  already  illustrated.  In  the  case  of  elliptic 
motion,  if  we  compute  the  auxiliary  quantities  e  and  d  by  means  of 
the  equations  (60),  we  shall  have 


e  cos  J  (E"  +  J£)  =  cos  J  (e  +  d), 

from  which  e  and  \(Efr -\~  E}  may  be  found,  and  hence,  since 
i(J£"—  J£)  =  i(e  —  £),  we  derive  .#  and  J0".  The  values  of  q  and 
i?  may  then  be  found  directly  from  these  and  quantities  already 
obtained.  Thus,  the  last  of  equations  (43)!  gives 

cos  ^v cos  \E  cos  ^v" cos  ^E" 

V~q    '       1/r  Vq  1/7' 

Multiplying  the  first  of  these  expressions  by  sin  Ju",  and  the  second 
oy  —  sin  J0,  adding  the  products,  and  reducing,  we  obtain 

smi(v"—  v)sin%v  _  cos  j  (v"  —  v)  cos  %E      cos  \E" 
l/q  Vr  V7'   ' 

Therefore,  we  shall  have 

_1_  g.n       _  cosjJg cos  ±E" 

V~q  Sm  3V  "  ~  1/r  tan  J  (M"—  *)       T/7'  sin  J  (t*"—  uj 
1  cos  IE 


-—=  cos  Iv  = -, 

VQ  V 


from  which  q  and  v  may  be  found  as  soon  as  cos  \E  and  cos  \E"  are 
known.  In  the  case  of  parabolic  motion  the  eccentric  anomaly  is 
equal  to  zero,  and  these  equations  become  identical  with  (92)3.  The 
angular  distance  of  the  perihelion  from  the  ascending  node  will  be 
obtained  from 

(a  =  u  —  V. 

Since  r  =  a  —  ae  co$E,  and  q  =  a(l  —  e),  we  have 


and  hence 


1-1 

a 


(89) 


352  THEORETICAL   ASTRONOMY. 

When  the  eccentricity  is  nearly  equal  to  unity,  the  value  of  q  given 
by  approximate  elements  will  be  sufficient  to  compute  cos^  and 
cos^E"  by  means  of  these  equations,  and  the  results  thus  derived 
will  be  substituted  in  the  equations  (88),  from  which  a  new  value  of 
q  results.  If  this  should  differ  considerably  from  that  used  in  com- 
puting cos  \E  and  cos  \E",  a  repetition  of  the  calculation  will  give 
the  correct  result. 

In  the  case  of  hyperbolic  motion,  although  E  and  Ef>  are  imagi- 
nary, we  may  compute  the  numerical  values  of  cos^  and  cos^E'' 
from  the  equations  (89),  regarding  a  as  negative,  and  the  results  will 
be  used  for  the  corresponding  quantities  in  (88)  in  the  computation 
of  q  and  v  for  the  hyperbolic  orbit. 

Next,  we  compute  a  second  system  of  elements  from  M and/ -f  Sf, 
and  a  third  system  from  M  -f-  dM  and/,  df  and  dM  denoting  the 
arbitrary  increments  assigned  to  /  and  M  respectively.  The  com- 
parison of  these  three  systems  of  elements  with  additional  observed 
places  of  the  comet,  will  enable  us  to  form  the  equations  of  condition 
for  the  determination  of  the  most  probable  values  of  the  corrections 
&M  and  A/ to  be  applied  to  M  and  /respectively.  The  formation  of 
these  equations  is  effected  in  precisely  the  same  manner  as  in  the  case 
of  the  variation  of  the  geocentric  distances  or  of  &  and  i,  and  it  does 
not  require  any  further  illustration.  The  final  elements  will  be  ob- 
tained from  M-\-  &M,  and/-f-  A/,  either  directly  or  by  interpolation. 
We  may  remark,  further,  that  it  will  be  convenient  to  use  log  M  as 
the  quantity  to  be  corrected,  and  to  express  the  variations  of  log  M 
in  units  of  the  last  decimal  place  of  the  logarithms. 

When  the  orbit  differs  very  little  from  the  parabolic  form,  it  will 
be  most  expeditious  to  make  two  hypotheses  in  regard  to  M,  putting 

in  each  case  —  =  0,  and  only  compute  elliptic  or  hyperbolic  elements 

in  the  third  hypothesis,  for  which  we  use  J/and  f=df.  The  first 
and  second  systems  of  elements  will  thus  be  parabolic. 

120.  Instead  of  M  and  -  we  may  use  A  and  -  as  the  quantities  to 
a  a 

be  corrected.  In  this  case  we  assume  an  approximate  value  of  J  by 
means  of  elements  already  known,  and  by  means  of  (96)3,  (98)3,  (102),, 
and  (103)3,  we  compute  the  auxiliary  quantities  C,  B,  B",  &c.,  re- 
quired in  the  solution  of  the  equations  (104)3.  We  assume,  also,  an 
approximate  value  of  Jr/  and  compute  the  corresponding  value  of  r", 
the  value  of  r  having  been  already  found  from  the  assumed  value  of 
J.  Then,  by  trial,  we  find  the  value  of  x  which,  in  connection  witn 


EQUATIONS   OF   CONDITION.  353 

the  assumed  value  of  -,  will  satisfy  the  equations  (104)3  and  (65)  or 
(61).  The  corresponding  value  of  A"  is  given  by 

J"  =  e  ±  i/x2  —  C\ 

When  A"  has  thus  been  determined,  the  heliocentric  places  will  be 
obtained  by  means  of  the  equations  (106)3  and  (107)3,  and,  finally, 
the  corresponding  elements  of  the  orbit  will  be  computed.  If  the 
ecliptic  is  taken  as  the  fundamental  plane,  we  put  D  =  Q,  A  =  O, 
and  write  ^  and  /9  in  place  of  a  and  d  respectively. 

If  we  now  compute  a  second  system  of  elements  from  A  +  d J  and 

f  =  -,  and  a  third  system  from  J  and  /+<?/",  the  comparison  of  the 

three  systems  of  elements  with  additional  observed  places  will  furnish 
the  equations  of  condition  for  the  determination  of  the  corrections 

A  J  and  A/  to  be  applied  to  J  and  -  respectively. 

When  the  eccentricity  is  very  nearly  equal  to  unity,  we  may  as- 
sume /=0  for  the  first  and  second  hypotheses,  and  only  compute 
elliptic  or  hyperbolic  elements  for  the  third  hypothesis. 

121.  The  comparison  of  the  several  observed  places  of  a  heavenly 
body  with  one  of  the  three  systems  of  elements  obtained  by  varying 
the  two  quantities  selected  for  correction,  or,  when  the  required  dif- 
ferential coefficients  are  known,  with  any  other  system  of  elements 
such  that  the  squares  and  products  of  the  corrections  may  be  neg- 
lected, gives  a  series  of  equations  of  the  form 

mx  -f  ny  —p, 
m'x  -f-  n'y  =p',  &c., 

in  which  x  and  y  denote  the  final  corrections  to  be  applied  to  the  two 
assumed  quantities  respectively.  The  combination  of  these  equations 
which  gives  the  most  probable  values  of  the  unknown  quantities,  is 
effected  according  to  the  method  of  least  squares.  Thus,  we  multiply 
each  equation  by  the  coefficient  of  x  in  that  equation,  and  the  sum 
of  all  the  equations  thus  formed  gives  the  first  normal  equation. 
Then  we  multiply  each  equation  of  condition  by  the  coefficient  of  y 
in  that  equation,  and  the  sum  of  all  the  products  gives  the  second 
normal  equation.  Let  these  equations  be  expressed  thus:-  — 

[mm]  x  +  [mn\  y  =  [mp], 

\mri\  x  +  \nri\  y  =  \np], 

23 


354  THEORETICAL   ASTRONOMY. 

in  which  [mra]=m2-{-m'2-|-m''2-}-&c.,  [mn]=mw+mV-f ra'W'-f&c., 
and  similarly  for  the  other  terms.  These  two  final  equations  give, 
by  elimination,  the  most  probable  values  of  x  and  y,  namely,  those 
for  which  the  sum  of  the  squares  of  the  residuals  will  be  a  minimum. 
It  is,  however,  often  convenient  to  determine  x  in  terms  of  y,  or  y 
in  terms  of  x,  so  that  we  may  find  the  influence  of  a  variation  of  one 
of  the  unknown  quantities  on  the  differences  between  computation 
and  observation  when  the  most  probable  value  of  the  other  unknown 
quantity  is  used.  Thus,  if  it  be  desired  to  find  x  in  terms  of  y,  the 
most  probable  value  of  x  will  be 


[mp~\         \mn\ 
[mm]        [mm] 


If  we  substitute  this  value  of  x  in  the  original  equations  of  condition, 
the  remaining  differences  between  computation  and  observation  will 
be  expressed  in  terms  of  the  unknown  quantity  y,  or  in  the  form 

A0  =  m0  +  n0y.  (90) 

Then,  by  assigning  different  values  to  y,  we  may  find  the  correspond- 
ing residuals,  and  thus  determine  to  what  extent  the  correction  y  may 
be  varied  without  causing  these  residuals  to  surpass  the  limits  of  the 
probable  errors  of  observation. 

In  the  determination  of  the  orbit  of  a  comet  there  must  be  more 
or  less  uncertainty  in  the  value  of  a,  and  if  y  denotes  the  correction 

to  be  applied  to  the  assumed  value  of  -,  we  may  thus  determine  the 

CL 

probable  limits  within  which  the  true  value  of  the  periodic  time 
must  be  found.  In  the  case  of  a  comet  which  is  identified,  by  the 
similarity  of  elements,  with  one  which  has  previously  appeared,  if 
we  compute  the  system  of  elements  which  will  best  satisfy  the  series 
of  observations,  the  supposition  being  made  that  the  comet  has  per- 
formed but  one  revolution  around  the  sun  during  the  intervening 
interval,  it  will  be  easy  to  determine  whether  the  observations  are 
better  satisfied  by  assuming  that  two  or  more  revolutions  have  been 
completed  during  this  interval.  Thus,  let  T  denote  the  periodic 
time  assumed,  and  the  relation  between  T  and  a  is  expressed  by 


k  ' 

in  which  n  denotes  the  semi-circumference  of  a  circle  whose  radius 


ORBIT   OF   A   COMET.  355 

is  unity.     Let  the  periodic  time  corresponding  to  -  -f-  y  be  denoted 

m     *  Q> 

by  —  ;  then  we  shall  have 

1   ,       1 
y  =  i*~7 

and  the  equations  for  the  residuals  are  transformed  into  the  form 

A    0  f  f\          I  f      \  f(\-\  \ 

A0  =  (m0  —  nj  )  -f-  n0j  z*.  (91) 

If  we  now  assign  to  2,  successively,  the  values  1,  2,  3,  &c.,  the  re- 
siduals thus  obtained  will  indicate  the  value  of  z  which  best  satisfies 
the  series  of  observations,  and  hence  how  many  revolutions  of  the 
comet  have  taken  place  during  the  interval  denoted  by  T. 

122.  In  the  determination  of  the  orbit  of  a  comet  from  three  ob- 
served places,  a  hypothesis  in  regard  to  the  semi-transverse  axis  may 
writh  facility  be  introduced  simultaneously  with  the  computation  of 
the  parabolic  elements.  The  numerical  calculation  as  far  as  the  form- 
ation of  the  equations  (52)s  will  be  precisely  the  same  for  both  the 
parabolic  and  the  elliptic  or  hyperbolic  elements.  Then  in  the  one 
case  we  find  the  values  of  r,  rf/,  and  x  which  will  satisfy  equation 
(56)3,  and  in  the  other  case  we  find  those  which  will  satisfy  the  equa- 
tion (65),  as  already  explained.  From  the  results  thus  obtained,  the 

two  systems  of  elements  will  be  computed.     Let  /=  ->  then  in  the 

case  of  the  system  of  parabolic  elements  Ave  have/=0,  and  the  com- 
parison of  the  middle  place  with  these  and  also  with  the  elliptic  or 
hyperbolic  elements  will  give  the  value  of 


in  which  6l  denotes  the  geocentric  spherical  co-ordinate  computed 
from  the  parabolic  elements,  and  02  that  computed  from  the  other 
system  of  elements.  Further,  let  A#  denote  the  difference  between 
computation  and  observation  for  the  middle  place,  and  the  correction 
to  be  applied  to  /,  in  order  that  the  computed  and  the  observed 
values  of  6  may  agree,  will  be  given  by 


Hence,  the  two  observed  spherical  co-ordinates  for  the  middle  place 
will  give  two  equations  of  condition  from  which  A/  may  be  found, 


356  THEORETICAL   ASTRONOMY. 

and  the  corresponding  elements  will  be  those  which  best  represent 
the  observations,  assuming  the  adopted  value  of  M  to  be  correct. 

123.  The  first  determination  of  the  approximate  elements  of  the 
orbit  of  a  comet  is  most  readily  effected  by  adopting  the  ecliptic  as 
the  fundamental  plane.  In  the  subsequent  correction  of  these  ele- 

ments, by  varying  -  and  M  or  J,  it  will  often  be  convenient  to  use 
a 

the  equator  as  the  fundamental  plane,  and  the  first  assumption  in 
regard  to  M  will  be  made  by  means  of  the  values  of  the  distances 
given  by  the  approximate  elements  already  known.  But  if  it  be 
desired  to  compute  M  directly  from  three  observed  places  in  reference 
to  the  equator,  without  converting  the  right  ascensions  and  declina- 
tions into  longitudes  and  latitudes,  the  requisite  formula  may  be 
derived  by  a  process  entirely  analogous  to  that  employed  when  the 
curtate  distances  refer  to  the  ecliptic.  The  case  may  occur  in  which 
only  the  right  ascension  for  the  middle  place  is  given,  so  that  the 
corresponding  longitude  cannot  be  found.  It  will  then  be  necessary 
to  adopt  the  equator  as  the  fundamental  plane  in  determining  a 
system  of  parabolic  elements  by  means  of  two  complete  observations 
and  this  incomplete  middle  place.  If  we  substitute  the  expressions 
for  the  heliocentric  co-ordinates  in  reference  to  the  equator  in  the 
equations  (4)3  and  (5)s,  we  shall  have 

0  —  n  (p  cos  a  —  R  cos  D  cos  A}  —  (pr  cos  a'  —  Rf  cos  U  cos  J/) 

-f  n"  (P"  sin  a"—  R"  cos  D"  cos  A"\ 

0  =  n  (p  sin  a  —  R  cos  D  sin  A)  —  (pf  sin  a'  —  E'  cos  D'  sin  A')  (92) 

-f  n"  (p"  sin  a"—  R"  cos  D"  sin  A"), 

Q  =  n(pttmd  —  R  sin  D}  —  (p  tan  S'—  R'  sin  Z>') 

-f-  n"  (p"  tan  a"  —  R"  sin  IT), 

in  which  py  //,  ptf  denote  the  curtate  distances  with  respect  to  the 
equator,  A,  Af,  A"  the  right  ascensions  of  the  sun,  and  D,  Z>',  I/' 
its  declinations.  These  equations  correspond  to  (6)3,  and  may  be 
treated  in  a  similar  manner. 

From  the  first  and  second  of  equations  (92)  we  get 

0  =  n  (p  sin  (a'—  a)  —RcosD  sin  (of—A))  -f  R  cos  D'  sin  (a'—  4') 

—  n"  (P"  sin  (a"—  a')  -f  R"  cosZ>"  sin  (a'—  A")), 
and  hence 


=?    =  ^>4na-a 
p         n"     Sin  (a"  —  a') 

?i  ft  cos  D  sin  (a'—  A)—  Rr  cosD'  sin  (a'-  A'}-\-ri'R"  cos  D"  sin  (a'—  A") 

pri'sm(a"—  a') 


VARIATION   OF   TWO   RADII- VECTORES.  357 

This  formula,  being  independent  of  the  decimation  d',  may  be  used 
to  compute  M  when  only  the  right  ascension  for  the  middle  place  ia 
given.  For  the  first  assumption  in  the  case  of  an  unknown  orbit, 
we  take 

1f     sin  (a'  —  a) 


M= 


if—  t     sin(a'/  —  a')' 


and,  by  means  of  the  results  obtained  from  this  hypothesis,  the  com- 
plete expression  (93)  may  be  computed.  By  a  process  identical  with- 
that  employed  in  deriving  the  equation  (36)3,  we  derive,  from  (93), 
the  expression 

p"  =  p£--j£$~$-)  (94) 

j  TT  f^\tt\i^-         \  \Rr  cos  D'  sin  (a'  —  A!} 
-g^F^       ^;\r*      #»/ sin  (a"  —  a') 
and,  putting 

,_.        n      sin  (a' — a) 


n"  '  sin  (a"  —  a') ' 

F—  I  —  '  —   rr*  (?'      T^  cos  D' sin  (a'  —  A')    ^/1__1_\ 
^         g  n  '  r"  ^T  sin  (a'  —  a)  p  \r'3      R'3f' 

we  have 

(95) 


The  calculation  of  the  auxiliary  quantities  in  the  equations  (52)3 
will  be  effected  by  means  of  the  formulae  (96)3,  (86),  (87),  (102)3,  and 
(51)3.  The  heliocentric  places  for  the  times  t  and  t"  will  be  given 
by  (106)3  and  (107)3,  and  from  these  the  elements  of  the  orbit  will 
be  found  according  to  the  process  already  illustrated. 

124.  The  methods  already  given  for  the  correction  of  the  approxi- 
mate elements  of  the  orbit  of  a  heavenly  body  by  means  of  additional 
observations  or  normal  places,  are  those  which  will  generally  be 
applied.  There  are,  however,  modifications  of  these  which  may  be 
advantageous  in  rare  and  special  cases,  and  which  will  readily  suggest 
themselves.  Thus,  if  it  be  desired  to  correct  approximate  elements 
by  varying  two  radii-vectores  r  and  r",  we  may  assume  an  approxi- 
mate value  of  each  of  these,  and  the  three  equations  (88)j  will  con- 
tain only  the  three  unknown  quantities  d,  b,  and  /.  By  elimination, 
these  unknown  quantities  may  be  found,  and  in  like  manner  the 


S58  THEORETICAL   ASTRONOMY. 

values  of  J",  b"j  and  I".  It  will  be  most  convenient  to  compute 
the  angles  ^  an<^  Vj  anc^  then  find  z  and  z"  from 

R  sin  *  „      R"  sin  V 

sm  z  =  -  ,  sm  z  =  -  ^  —  , 

or,  putting  #2  =  r2  —  J?2  sin2^,  and  x"2  =  r"2  —  -R"2  sin2^",  from 

£  sin  4  „      £"  sin  V 

tan  z  =  -  ,  tan  z  =  -  „  —  . 

x  x" 

The  curtate  distances  will  be  given  by  the  equations  (3),  and  the 
heliocentric  spherical  co-ordinates  by  means  of  (4),  writing  r  in  place 
of  a.  From  these  u"  —  u  may  be  found,  and  by  means  of  the  values 
of  r,  rn  ',  and  u"  —  u  the  determination  of  the  elements  of  the  orbit 
may  be  completed.  Then,  assigning  to  r  an  increment  dr,  we  com- 
pute a  second  system  of  elements,  and  from  r  and  r"  -f-  drff  a  third 
system.  The  comparison  of  these  three  systems  of  elements  with  an 
additional  or  intermediate  observed  place  will  furnish  the  equations 
for  the  determination  of  the  corrections  Ar  and  Ar/r  to  be  applied  to 
r  and  r",  respectively.  The  comparison  of  the  middle  place  may  be 
made  with  the  observed  geocentric  spherical  co-ordinates  directly,  or 
with  the  radius-vector  and  argument  of  the  latitude  computed  directly 
from  the  observed  co-ordinates;  and  in  the  same  manner  any  number 
of  additional  observed  places  may  be  employed  in  forming  the  equa- 
tions of  condition  for  the  determination  of  Ar  and  Arr/. 

Instead  of  r  and  r",  we  may  take  the  projections  of  these  radii- 
vectores  on  the  plane  of  the  ecliptic  as  the  quantities  to  be  corrected. 
Let  these  projected  distances  of  the  body  from  the  sun  be  denoted 
by  r0  and  r0",  respectively  ;  then,  by  means  of  the  equations  (88)^ 
we  obtain 


(96) 


from  which  I  may  be  found  ;  and  in  a  similar  manner  we  may  find 
I".     If  we  put 


we  have 

tan(;-,>)  =  *sin(A-0).  (97, 

XQ 

Let  8  denote  the  angle  at  the  sun  between  the  earth  and  the  place 
of  the  planet  or  comet  projected  on  the  plane  of  the  ecliptic  ;  then 
we  shall  have 


VARIATION   OF   TWO   RADII-VECTORES.  359 

£=180°  +  O—  I, 
==:R8in(Z-L0)  (98) 

and 

p  tan  ft 

tan  6  = ,  (99) 

ro 

by  means  of  which  the  heliocentric  latitudes  b  and  b"  may  be  found. 
The  calculation  of  the  elements  and  the  correction  of  r0  and  r0"  are 
then  effected  as  in  the  case  of  the  variation  of  r  and  r" . 

In  the  case  of  parabolic  motion,  the  eccentricity  being  known,  we 
may  take  q  and  T  as  the  quantities  to  be  corrected.  If  we  assume 
approximate  values  of  these  elements,  r,  r',  rff,  and  v,  vf,  v"  will  be 
given  immediately.  Then  from  r,  rr,  r'1  and  the  observed  spherical 
co-ordinates  of  the  body  we  may  compute  the  values  of  u"  —  u'  and 
u'  —  u.  In  the  same  manner,  by  means  of  the  observed  places,  wo 
compute  the  angles  u" — u'  and  uf — u  corresponding  to  q-\-dq  and  Ty 
and  to  q  and  T-+-  d  T}  3q  and  dT  denoting  the  arbitrary  increments 
assigned  to  q  and  T,  respectively.  The  comparison  of  the  helio- 
centric motion,  during  the  intervals  t"  —  t1  and  t'  —  i,  thus  obtained, 
in  the  case  of  each  of  the  three  systems  of  elements,  from  the  ob- 
served geocentric  places  with  the  corresponding  results  given  by 

w"  —  u'  =  v"  —  v', 

enables  us  to  form  the  equations  by  which  we  may  find  the  cor- 
rections A^  and  &T  to  be  applied  to  the  assumed  values  of  q  and  T, 
respectively,  in  order  that  the  values  of  u"  —  uf  and  u'  —  u  computed 
by  means  of  the  observed  places  shall  agree  with  those  given  by  the 
true  anomalies  computed  directly  from  q  and  T. 


360  THEORETICAL   ASTRONOMY. 


CHAPTER  VII. 

METHOD  OP  LEAST  SQUARES,  THEORY  OF  THE  COMBINATION  OP  OBSERVATIONS,  AND 
DETERMINATION  OF  THE  MOST  PROBABLE  SYSTEM  OF  ELEMENTS  FROM  A  SERIES 
OF  OBSERVATIONS. 

125.  WHEN  the  elements  of  the  orbit  of  a  heavenly  body  are  known 
to  such  a  degree  of  approximation  that  the  squares  and  products  of 
the  corrections  which  should  be  applied  to  them  may  be  neglected, 
by  computing  the  partial  differential  coefficients  of  these  elements 
with  respect  to  each  of  the  observed  spherical  co-ordinates,  we  may 
form,  by  means  of  the  differences  between  computation  and  observa- 
tion, the  equations  for  the  determination  of  these  corrections.  Three 
complete  observations  will  furnish  the  six  equations  required  for  the 
determination  of  the  corrections  to  be  applied  to  the  six  elements  of 
the  orbit ;  but,  if  more  than  three  complete  places  are  given,  the 
number  of  equations  will  exceed  the  number  of  unknown  quantities, 
and  the  problem  will  be  more  than  determinate.  If  the  observed 
places  were  absolutely  exact,  the  combination  of  the  equations  of 
condition  in  any  manner  whatever  would  furnish  the  values  of  these 
corrections,  such  that  each  of  these  equations  would  be  completely 
satisfied.  The  conditions,  however,  which  present  themselves  in  the 
actual  correction  of  the  elements  of  the  orbit  of  a  heavenly  body  by 
means  of  given  observed  places,  are  entirely  different.  When  the 
observations  have  been  corrected  for  all  known  instrumental  errors, 
and  when  all  other  known  corrections  have  been  duly  applied,  there 
still  remain  those  accidental  errors  which  arise  from  various  causes, 
such  as  the  abnormal  condition  of  the  atmosphere,  the  imperfections 
of  vision,  and  the  imperfections  in  the  performance  of  the  instrument 
employed.  These  accidental  and  irregular  errors  of  observation  cannot 
be  eliminated  from  the  observed  data,  and  the  equations  of  condition 
for  the  determination  of  the  corrections  to  be  applied  to  the  elements 
of  an  approximate  orbit  cannot  be  completely  satisfied  by  any  system 
of  values  assigned  to  the  unknown  quantities  unless  the  number  of 
equations  is  the  same  as  the  number  of  these  unknown  quantities. 
It  becomes  an  important  problem,  therefore,  to  determine  the  par- 
ticular combination  of  these  equations  of  condition,  by  means  of  which 


METHOD   OF   LEAST   SQUARES.  361 

the  resulting  values  of  the  unknown  quantities  will  be  those  which, 
while  they  do  not  completely  satisfy  the  several  equations,  will  afford 
the  highest  degree  of  probability  in  favor  of  their  accuracy.  It  will 
be  of  interest  also  to  determine,  as  far  as  it  may  be  possible,  the 
degree  of  accuracy  which  may  be  attributed  to  the  separate  results. 
But,  in  order  to  simplify  the  more  general  problem,  in  which  the 
quantities  sought  are  determined  indirectly  by  observation,  it  will  be 
expedient  to  consider  first  the  simpler  case,  in  which  a  single  quantity 
is  obtained  directly  by  observation. 

126.  If  the  accidental  errors  of  observation  could  be  obviated,  the 
different  determinations  of  a  magnitude  directly  by  observation  would 
be  identical ;  but  since  this  is  impossible  when  an  extreme  limit  of 
precision  is  sought,  we  adopt  a  mean  or  average  value  to  be  derived 
from  the  separate  results  obtained.  The  adopted  value  may  or  may 
not  agree  with  any  individual  result,  since  it  is  only  necessary  that 
the  residuals  obtained  by  comparing  the  adopted  value  with  the 
observed  values  shall  be  such  as  to  make  this  adopted  value  the  most 
probable  value.  It  is  evident,  from  the  very  nature  of  the  case,  that 
we  approach  here  the  confines  of  the  unknown,  and,  before  we  pro- 
ceed further,  something  additional  must  be  assumed. 

However  irregular  and  uncertain  the  law  of  the  accidental  errors 
of  observation  may  be,  we  may  at  least  assume  that  small  errors  are 
more  probable  than  large  errors,  and  that  errors  surpassing  a  certain 
limit  will  not  occur.  We  may  also  assume  that  in  the  case  of  a  large 
number  of  observations,  errors  in  excess  will  occur  as  frequently  as 
errors  in  defect,  so  that,  in  general,  positive  and  negative  residuals 
of  equal  absolute  value  are  equally  probable.  It  appears,  therefore, 
that  the  relative  frequency  of  the  occurrence  of  an  accidental  error  A 
in  the  observed  value  will  depend  on  the  magnitude  of  this  error, 
and  may  be  expressed  by  <p  (A).  This  function  will  also  express  the 
probability  of  an  error  A  in  an  observed  value.  At  the  limit  beyond 
which  an  error  of  the  magnitude  A  can  never  occur,  we  must  have 
<p(A)  =  0:  when  A  =  0,  the  value  of  <p  (A)  must  be  a  maximum,  and 
for  equal  positive  and  negative  values  of  A  the  values  of  <p  (A)  must 
be  the  same.  Hence,  in  a  given  series  of  observations,  the  number  m 
of  observations  being  supposed  to  be  large,  the  number  of  times  in 
which  the  error  A  occurs  will  be  expressed  by  m<p  ( J),  and  the  number 
of  times  in  which  the  error  A'  occurs  will  be  expressed  by  m<p  (J1),  so 
that  we  shall  have 

-f  my  (J")  -\-  &c., 


362  THEORETICAL   ASTRONOMY. 

or 


The  sum  2  must  be  taken  between  the  limits  for  which  the  accidental 
errors  of  observation  are  considered  possible  ;  but  since  the  assignment 
of  these  limits  is,  in  a  certain  sense,  arbitrary,  we  must  evidently 
have 

A=  -foo 

?W  =  1,  (1) 


the  value  of  <p  (  A)  being  absolutely  zero  for  the  limits  -4-  oo  and  —  oo. 
Within  any  given  limits  there  are  an  infinite  number  of  values, 
any  one  of  which  may  possibly  be  the  true  value  of  J,  and  hence 
the  number  of  the  functions  expressed  by  <p  (A)  must  be  infinite. 
The  probability  of  an  error  A  is  expressed  by  <p  (  J),  and  will  be  the 
same  as  the  probability  that  the  error  is  contained  within  the  limits  A 
and  A  -f-  dA.  The  latter  is  expressed  by  the  sum  of  all  the  functions 
(p  (  A)  between  the  limits  A  and  A  -f-  dJ,  or  by 


We  conclude,  therefore,  that  the  probability  that  an  error  falls  between 
the  limits  a  and  b  is  expressed  by  the  integral 


and  this  integral,  taken  so  as  to  include  all  possible  accidental  errors 
of  observation,  is,  according  to  equation  (1), 


According  to  the  theory  of  probabilities,  the  probability  that  the 
errors  J,  Jr,  &c.  occur  simultaneously  is  equal  to  the  continued  pro- 
duct of  the  probabilities  of  the  occurrence  of  these  errors  separately. 
Let  P  denote  the  probability  that  these  errors  occur  at  the  same  time 
in  the  given  series  of  observed  values,  and  we  have 

p=?(j).?(j').?(.r)  .....  (3) 

The  most  probable  value  of  the  quantity  sought,  which  we  will  de- 
note by  x,  must  evidently  be  that  which  makes  P  a  maximum.     If 


METHOD   OF   LEAST   SQUARES.  363 

we  take  the  logarithms  of  both  members  of  equation  (3),  and  differ- 
entiate, the  condition  of  a  maximum  gives 


n  dJ   , 

-dT-'Tx  +  --dA—'-dx-  +  &c' 

Let  n,  nf,  n",  &c.  be  the  observed  values  of  x,  and  m  the  number  of 
observations  ;  then  we  have 


and  hence 

U_dX___dAT_ 
dx        dx        dx 

Therefore  the  equation  (4)  becomes 

d  log  y  (n  —  x)       d  log  <?  (ri  —  x) 

d(n-x)  d(n'-x)    -  +  &C'  (5) 

This  equation  will  serve  to  determine  the  value  of  x  as  soon  as  the 
form  of  the  function  symbolized  by  <p  is  known.  It  becomes  neces- 
sary, therefore,  to  make  some  further  assumption  in  regard  to  the 
errors  J,  J',  A",  &c.,  in  order  that  the  form  of  this  function  may  be 
determined;  and,  although  the  hypothesis  which  presents  itself  gives 
directly  the  most  probable  value  of  x,  since  the  function  <p  (  J)  is  sup- 
posed to  be  general,  we  may  thus,  by  the  special  case,  determine  the 
form  of  this  function;  and  the  result  will  be  applicable  when,  instead 
of  the  value  of  a  single  quantity,  it  is  required  to  find  the  most  pro- 
bable values  of  several  unknown  quantities  determined  indirectly  by 
observation. 

127.  The  principle  may  be  received  as  an  axiom,  that  when  a 
series  of  observed  values  of  a  quantity  is  given,  if  the  circumstances 
under  which  the  separate  observations  were  made  are  similar,  so  that 
there  is  no  reason  for  preferring  one  result  to  another,  the  most  pro- 
bable value  of  the  quantity  sought  is  the  arithmetical  mean  of  the 
several  results.  Hence  we  have 


x  = 


m 
m  being  the  number  of  observed  values.     This  expression  gives 

0  =  (n  —  x)  -f  (n1  —  x}  +  (n"  —  x)  -f-  &c.,  (6) 

from  which  it  appears  that  the  algebraic  sum  of  the  residuals  is  equal 
to  zero.     The  equation  (5)  may  be  written 


364  THEORETICAL   ASTRONOMY. 

d  log  <f  (n  —  x)  ,.     d  log  (f>  (n'  —  x) 

Q  =  (n  —  x)  7  -  ^rjr,  --  ^  +  (ri  —  x)  7-?  —  &\\,  ,  —  —  ,  4-  &c., 
(n  —  x)  d  (n  —  x)  (ri  —  x)  d  (ri  —  x)   ' 

and  the  comparison  of  this  with  (6)  shows  that 

d  log  <p  (n  —  a?)  d  log  <f>  (ri  —  x) 

(n  —  x)d(n  —  x)       (ri  —x)d(ri  —  x)  " 

k  being  a  constant  quantity.     Hence  we  derive 

d  loge  <?  (  J)  =  kA  d  J, 
the  integration  of  which  gives 


loge  c  being  the  constant  of  integration.  From  this  equation  tiiere 
results 

,    ,,  ^*A»  , 

^  (  J)  =  ce      ,  (8) 

in  which  e  is  the  base  of  Naperian  logarithms.  Since  <p  (  J)  diminishes 
as  A  increases,  the  quantity  k  must  be  essentially  negative,  and  if  we 
put  %k  =  —  A2,  we  shall  have 


c<f  (9) 

If  we  substitute  this  value  of  (p(A)  in  the  equation  (2),  we  have 


00 

/•    -A3  A' 

c  J  e         dA  =  1, 
or,  putting  also  t  =  liA, 


c 

-  00 


This  equation  will  give  the  value  of  the  constant  c,  provided  that  the 

value  of  the  integral 

/»°° 

f  •-** 

»/  o 

is  known.  Since  the  definite  integral  is  independent  of  the  variable, 
let  us  multiply  it  by  a  similar  one,  in  which  y  is  the  variable  ;  so 
that  we  have 


ID  which  the  order  of  integration  is  indifferent.     If  we  put  y  =  te, 


METHOD   OF   LEAST   SQUARES.  365 

we  have,  since  t  is  regarded  as  constant  in  the  integration  with  respect 

toy, 

dy  =  tdz  ; 
and  hence 


Then,  since  we  have,  in  general, 


1 

7r, 
2a 


the  preceding  equation  gives 


in  which  n  denotes  the  semi-circumference  of  a  circle  whose  radius  is 
unity.  Therefore  we  have 

fa-*  eft  =  £!/£;  (11) 

*/  o 

and  the  equation  (10)  gives 

e  =  4.  (12) 

I/* 

Hence,  the  expression  for  <p  (A]  becomes 

r(J)=A«-«».  (13) 

V* 

The  constant  h,  according  to  the  relation  tf  =  —  \k,  must  depend  on 
the  nature  of  the  observations,  and  will  be  the  same  in  the  case  of 
systems  of  observations  in  which  the  probability  of  an  error  A  is  the 

same.  Since  A2J2  must  necessarily  be  an  abstract  number,  A  and  ^ 
must  be  homogeneous. 

128.  In  a  given  series  of  observations,  the  probability  that  for  any 
observation  the  error  will  be  within  the  limits  —  d  and  -f  d  will  be 
expressed  by 

+  5 

JL   fe-A'A'dJ;  (14) 

l/?rJ 
—  S 

and  in  another  series  of  observations,  more  or  less  precise,  the  pro- 


366  THEORETICAL   ASTRONOMY. 

bability  that  the  error  of  an  observation  is  within  the  limits  —  3'  and 
+  3'  will  be 

r^dA.  (is) 

-5' 

Since 

+  8  +hS 


=  -i=   f  e-A9A' 

i/-J 


it  appears  that  the  integrals  (14)  and  (15)  are  equal  when  h3  =  h'3f. 
Hence,  if  we  put  hf  =  2h,  these  integrals  will  be  equal  when  3  =  23', 
and  an  error  of  a  given  magnitude  in  the  first  series  will  have  the 
same  probability  as  an  error  of  half  that  magnitude  in  the  second 
series.  The  second  series  of  observations  will  therefore  be  twice  as 
accurate  as  the  first  series,  and  the  constant  h  may  be  called  the 
measure  of  precision  of  the  observations.  The  greater  the  degree  of 
precision  of  the  observations,  the  greater  will  be  the  value  of  h. 

The  relative  accuracy  of  two  series  of  observations  may  also  be 
determined  by  a  comparison  of  the  errors  which  are  committed  with 
equal  facility  in  each  series.  If  we  arrange  the  errors  of  the  several 
observations  in  each  series  in  the  order  of  their  absolute  magnitude 
without  reference  to  the  algebraic  sign,  the  errors  which  occupy  the 
same  position  in  reference  to  the  extremes  in  each  case  will  serve  to 
determine  the  relation  sought.  We  select  that,  however,  which  occu- 
pies the  middle  place  in  the  series  of  errors  thus  arranged,  and  since 
the  number  of  errors  which  exceed  this  is  the  same  as  the  number 
of  errors  less  than  this,  if  we  designate  the  error  which  occupies  the 
middle  place  by  r,  the  probability  that  an  error  is  within  the  limits 
—  r  and  +  f  will  be  equal  to  \.  The  probability  of  an  error  greater 
than  r  being  the  same  as  the  probability  of  an  error  less  than  r,  the 
error  r  is  called  the  probable  error. 

The  relation  between  r  and  h  is  easily  determined.     Thus,  we  have 


-f 

i/w 


or,  putting  h  A  =  t, 

Ht 

dt  =  ^  =  0.44311.  (16) 


f 

t/  o 


If  we  expand  e~**  into  a  series  of  ascending  powers  of  t,  multiply  by 
dty  and  integrate  between  the  limits  0  and  T,  we  get 


METHOD   OF  LEAST  SQUARES.  367 

T 
/*  npo  rni  rp9 

J  n  ^  '      s>  1        O  ^  1      2      S      '     ^  1       2      ^J      4  *' 


which  converges  rapidly  when  T  is  small.  To  find  the  value  of  T 
which  corresponds  to  the  value  0.44311  assigned  to  the  integral,  we 
compute  the  value  of  the  series  (17)  for  the  values  0.45,  0.47,  and 
0.49  assigned  to  T,  successively,  and  from  the  results  thus  obtained 
it  is  easily  seen  that  when  the  sum  of  the  terms  of  the  series  is 

0.44311,  we  have 

T=Jir  =  0.47694, 
or 

0.47694 
r  =  — -j— ,  (18) 

which  determines  the  relation  between  the  probable  error  and  the 
measure  of  precision. 

The  probability  that  the  error  of  an  observation,  without  regard  to 
sign,  does  not  exceed  nr,  is  expressed  by 


,— >        I        <-/  wi/j  (  J.t7  ) 

VTT^O 

and  this  integral,  therefore,  indicates  the  ratio  of  the  number  of  obser- 
vations affected  with  an  error  which  does  not  exceed  nr  to  the  whole 
number  of  observations.  Hence,  if  we  assign  different  values  to  n, 
the  integral  (19)  computed  for  the  several  assumed  values  of 

nhr  =  0.47694n 

will  give  the  relative  number  of  errors  of  a  given  magnitude.  Thus, 
if  we  put  n  =  \)  we  obtain 

0.2385 

-4:  JV"  eft  =  0.264. 

from  which  it  appears  that  in  a  series  of  1000  observations  there 
ought  to  be  264  observations  in  which  the  error  does  not  exceed  \r. 
It  has  been  found,  in  this  manner,  that  in  the  case  of  an  extended 
series  of  observations  the  number  of  errors  of  a  given  magnitude 
assigned  by  theory  agrees  very  closely  with  that  actually  given  by 
the  series  of  observations ;  and  hence  we  conclude  that  the  error  com- 
mitted in  extending  the  limits  of  the  summation  in  the  expression  (1) 
to  —  oo  and  -f  oo,  instead  of  the  finite  limits  which  it  is  presumed 
that  the  actual  errors  cannot  exceed,  is  very  slight,  so  that  the  form 


368  THEORETICAL   ASTRONOMY. 

of  the  function  (p  (J)  which  has  been  derived  may  be  regarded  as  that 
which  best  satisfies  all  the  conditions  of  the  problem. 

129.  The  relative  accuracy  of  different  series  of  observations  may 
also  be  indicated  by  means  of  what  are  called  the  mean  error  and  the 
mean  of  the  errors  for  each  series,  the  former  being  the  error  whose 
square  is  equal  to  the  mean  of  the  squares  of  all  the  errors  of  the 
series,  and  the  latter  the  mean  of  these  errors  without  reference  to 
their  algebraic  sign. 

Let  £  denote  the  mean  error ;  then,  since  the  number  of  observa- 
tions having  the  error  J  is  m<p  ( J),  we  shall  have,  according  to  the 
definition, 


m 


But  the  number  of  possible  errors  being  infinite,  the  probability  of 
an  error  J  is  expressed  by  <p  (  A)  dd,  and  we  have 


=  f  JV  (  J)  dA  =  A  f  e- 


which  gives 


Hence,  by  means  of  (18),  we  have 

e  =  -V  —  1.4826r, 

AT/2  (21) 

r=  0.67449s, 

which  determine  the  relation  between  e  and  r. 

Let  r}  denote  the  mean  of  the  errors,  and  we  shall  have 

/*  01        /'a° 

1A<p(A)dA  =  -  -   I    (T^'JdJ, 
-J//7T  ^  0 

which  gives 


Therefore,  we  have 

T)  =  1.1829r, 
r  =  0.8453^, 

for  the  relation  between  r  and  3. 


METHOD   OF   LEAST  SQUAKES.  369 

130.  Let  us  denote  by  v,  v',  v",  &c.  the  differences  between  any- 
assumed  value  of  x  and  the  observed  values  for  a  given  series  of 
observations,  the  number  of  observations  being  denoted  by  m;  then 
if  we  put 

[w]  =  v*  +  v'2  +  vn*  +  Ac.,  (24) 

and  similarly  in  the  case  of  the  sum  of  any  other  series  of  similar 
terms,  we  shall  have  for  the  probability  of  the  value  x,, 

P—  .       fr*       r~»W  /OCN 

~V* 

and  this  probability  will  be  a  maximum  when  [wi]  is  a  minimum 
Now  we  have 

o  —  n  —  xn  vr  =  n'  —  xn  v"  =  n"  —  xn  &c., 

w,  n',  n"j  &c.  being  the  observed  values  of  x,  and  hence 

[wj]  =  \nri]  —  2  [n]  x,  -(-  mxf* 

M\* 
»)• 

It  appears,  therefore,  that  [vv]  will  be  a  minimum  when 

*,  =  ™  (26) 

and  this  is  a  necessary  consequence  of  the  assumption  that  the  arith- 
metical mean  of  the  observations  gives  the  most  probable  value  of  x, 
according  to  which  the  form  of  the  function  <p  (  A]  was  derived.  But 
although  the  arithmetical  mean  is  the  most  probable  value,  yet  we 
cannot  affirm  that  this  is  the  exact  value,  so  long  as  the  number  of 
observations  is  finite.  It  becomes  important,  therefore,  to  determine 
the  degree  of  precision  of  the  arithmetical  mean. 

Let  XQ  denote  the  most  probable  value  of  x,  for  which  the  residuals 
are  v,  v',  v"j  &c.,  and  let  x0  +  d  be  any  other  value  of  #.  Then,  since 
we  may  put 

Wa=v-f.«r  +  ^-j.i.;.«o, 
and 

[vy]  =  me*, 

the  probability  of  the  value  x0  ~f-  d  will  be 


24 


370  THEORETICAL   ASTRONOMY. 

The  probability  that  the  error  of  the  arithmetical  mean  is  zero  is  in- 
dicated by 

P—  _^L_  ,-«**• 

*  —    /  —  e        > 
V  xm 

and  we  have 


In  the  case  of  a  single  observation,  if  P  denotes  the  probability  of 
the  error  zero,  and  Pf  the  probability  of  the  error  d,  we  have 


Hence  it  appears  that  if  h0  denotes  the  measure  of  precision  of  the 
arithmetical  mean  of  m  observations,  the  relation  between  h0  and  A, 
the  measure  of  precision  of  an  observation,  is  given  by 

h0*  =  mh*;  (27) 

and  if  r0  is  the  probable  error  of  the  arithmetical  mean,  and  £0  its 
mean  error,  we  have,  according  to  the  equations  (18)  and  (20), 


(28) 


These  expressions  determine  the  probable  and  the  mean  error  of  the 
arithmetical  mean  of  a  number  of  observations  when  these  errors  in 
the  case  of  a  single  observation  are  known. 

131.  The  expressions  for  the  relation  between  the  mean  and  pro- 
bable errors  have  been  derived  for  the  case  of  a  very  large  number 
of  observations,  a  number  so  great  that  the  error  of  the  arithmetical 
mean  becomes  equal  to  zero.  In  the  case  of  a  limited  number  of 
observed  values  of  #,  the  residuals  given  by  comparing  the  arith- 
metical mean  with  the  several  observations  will  not,  in  general,  give 
the  true  errors  of  the  observations ;  but  the  greater  the  number  of 
observations,  the  nearer  will  these  residuals  approach  the  absolute 
errors.  If  J,  Jr,  An ',  &c.  are  the  actual  errors  of  the  observations, 
and  v,  vf,  vff,  &c.  those  which  result  from  the  most  probable  value  of 
x,  we  shall  have,  denoting  the  arithmetical  mean  by  xw  and  the  true 
value  by  XQ  -f  3, 

A=v  —  S,  A'  =  v'  —  3,  A"  =  v"  —  d,  &c.; 


METHOD   OF  LEAST  SQUABES.  371 

hence 

me2  =  [  A  J]  =  [W]  +  m<*2.  (29) 

This  equation  will  enable  us  to  determine  the  mean  error  of  an  ob- 
servation when  3  is  given  ;  but,  since  this  is  necessarily  unknown, 
some  assumption  in  regard  to  its  value  must  be  made.  If  we  assume 
it  to  be  equal  to  the  mean  error  of  the  arithmetical  mean,  the  re- 
maining error  will  be  wholly  insensible,  and  hence  the  equation  (29) 
becomes 

we2  =  [iw]  +  me02  =  [vv~\  -f-  «3. 

Therefore,  we  shall  have 

< 

and,  according  to  (21), 

r  =  0.6745 


These  equations  give  the  values  of  the  mean  and  probable  errors  of 
a  single  .observation  in  terms  of  the  actual  residuals  found  by  com- 
paring the  arithmetical  mean  with  the  several  observed  values. 

The  probable  and  the  mean  error  of  the  arithmetical  mean  will  be 
given  by 


'  ^ (32; 

r0  =  0.6745^      |^j_     . 

When  the  number  of  observations  is  very  large,  the  probable  error 
of  an  observation  and  also  that  of  the  arithmetical  mean  may  be  de- 
termined by  means  of  the  mean  of  the  errors.  If  we  suppose  the 
number  of  positive  errors  to  be  the  same  as  the  number  of  negative 
errors,  the  mean  of  the  errors  without  reference  to  the  algebraic  sign 
gives 


and  hence  we  have,  according  to  (23), 

r  =  0.8453  &i  (33) 

m 

For  the  mean  error  of  an  observation  we  have 

e  =  »l/ii  =  1.2533  £4  (34) 


372  THEORETICAL  ASTRONOMY. 

If  the  number  of  observations  is  very  great,  the  results  given  oy 
these  equations  will  agree  with  those  given  by  (30)  and  (31);  but  for 
any  limited  series  of  observed  values,  the  results  obtained  by  means 
of  the  mean  error  will  afford  the  greatest  accuracy. 

132.  The  relative  accuracy  of  two  or  more  observed  values  of  a 
quantity  may  be  expressed  by  means  of  what  are  called  their  weights. 
If  the  observations  are  made  under  precisely  similar  circumstances, 
so  that  there  is  no  reason  for  preferring  one  to  the  other,  they  are  said 
to  have  the  same  weight.  The  weight  must  therefore  depend  on  the 
measure  of  precision  of  the  observations,  and  hence  on  their  probable 
errors.  The  unit  of  the  weight  is  entirely  arbitrary,  since  only  the 
relative  weights  are  required,  and  if  we  denote  the  weight  by  p,  the 
value  of  p  indicates  the  number  of  observations  of  equal  accuracy 
which  must  be  combined  in  order  that  their  arithmetical  mean  may 
have  the  same  degree  of  precision  as  the  observation  whose  weight  is 
p.  Hence,  if  the  weight  of  a  single  observation  is  1,  the  arithmetical 
mean  of  m  such  observations  will  have  the  weight  m.  Let  the  pro- 
bable error  of  an  observation  of  the  weight  unity  be  denoted  by  r, 
and  the  probable  error  of  that  whose  weight  is  pf  by  r'  •  then,  ac- 
cording to  the  first  of  equations  (28),  we  shall  have 


or 


For  the  case  of  an  observation  whose  weight  is  prr  and  whose  pro- 
bable error  is  r",  we  have 


from  which  it  appears  that  the  weights  of  two  observations  are  to  each 
other  inversely  as  the  squares  of  their  probable  or  mean  errors,  andf 
according  to  (18),  directly  as  the  squares  of  their  measures  of  precision. 
Let  us  now  consider  two  values  of  #,  which  may  be  designated  by 
xr  and  a?",  the  mean  errors  of  these  values  being,  respectively,  e'  and 

e":  then,  if  we  put 

X=x'±x" 

and  suppose  that  both  a;'  and  xtr  have  been  derived  from  a  large  num- 
ber m  of  observations  (and  the  same  number  in  each  case),  so  that  the 
residuals  v9  v/,  vn  ',  &c.  in  the  case  of  xf  and  the  residuals  v,,  v/,  v/'t 
&c.  in  the  case  of  xn  may  be  regarded  as  the  actual  errors  of  obser- 


METHOD   OF   LEAST  SQUARES. 

vatiou,  the  errors  of  the  value  of  Xt  as  determined  from  the  several 
observations,  will  be 

V  ±  V,,  V'  ±  V,',  V"  ±  V,",  &G. 

Let  the  mean  error  of  X  be  denoted  by  E;  then  we  have 

mE*  =S(v±  v,y  =  [w]  zh  2  [>,]  +  [v,vt]  ; 

and  since  the  number  of  observed  values  is  supposed  to  be  so  great 
that  the  frequency  of  negative  products  w,  is  the  same  as  that  of  the 
similar  positive  products,  so  that  [vvj]  =  0,  this  equation  gives 


or 

E2  =  e"  -I-  e"2. 

Combining  X  with  a  third  value  x"r  whose  mean  error  is  e'",  the 
mean  error  of  xf  ±  x"  ±  x1"  will  be  found  in  the  same  manner  to  be 
equal  to  e/2+  e//2-j-  s///2;  and  hence  we  have,  for  the  algebraic  sum 
of  any  number  of  separate  values, 


E  =     *  +  e'2  +  e"2  +  Ac.,  (35) 

and,  according  to  the  last  of  equations  (21), 


R  =  ir2-hr'2  +  r"2  +  &c.,  (36) 

R  being  the  probable  error  of  the  algebraic  sum.     If  the  probable 
errors  of  the  several  values  are  the  same,  we  have 


and  the  probable  error  of  the  sum  of  m  values  will  be  given  by 

E  =  rl/m. 

Hence  tne  probable  error  of  the  arithmetical  mean  of  m  observed 

values  will  be 

R         r 
*•=  —  =77—  » 

m       Vm 

which  agrees  with  the  first  of  equations  (28). 

Let  P  denote  the  weight  of  the  sum  X,  pf  the  weight  of  a?',  and  p/f 
that  of  x"  ;  then  we  shall  have 

p»  _ 

" 


374  THEORETICAL   ASTRONOMY. 

from  which  we  get 


Since  the  unit  of  weight  is  arbitrary,  we  may  take 


and  hence  we  have,  for  the  weight  of  the  algebraic  sum  of  any 
number  of  values, 

=  ^==t/1+  r"2-f-r'"2-{-&c.' 
or,  whatever  may  be  the  unit  of  weight  adopted, 


±.     I        •*•        I       ±  __  I 
p'     ""   p"      '     p'"     ' 

In  the  case  of  a  series  of  observed  values  of  a  quantity,  if  we 
designate  by  rf  the  probable  error  of  a  residual  found  by  comparing 
the  arithmetical  mean  with  an  observed  value,  by  r  the  probable 
error  of  the  observation,  by  XQ  the  arithmetical  mean,  and  by  n  any 
observed  value,  the  probable  error  of 

n=x0+v, 
according  to  (36),  will  be 

«•—•.•+**=;  +  •", 

r0  being  the  probable  error  of  the  arithmetical  mean.   Hence  we  derive 


m 


m  —  1 
and  if  we  adopt  the  value 

/  =  0.8453  ^3; 
m' 

the  expression  for  the  probable  error  of  an  observation  becomes 

r  =  0.8453          M  (40) 

l/m(m  —  1) 

in  which  [v]  denotes  the  sum  of  the  residuals  regarded  as  positive, 
and  m  the  number  of  observations. 

133.  Let  n,  nf,  n",  &c.  denote  the  observed  values  of  x,  and  let  p, 
p',  pff}  &c.  be  their  respective  weights  ;  then,  according  to  the  defi- 


METHOD   OF   LEAST  SQUARES.  375 

nition  of  the  weight,  the  value  n  may  be  regarded  as  the  arithmetical 
mean  of  p  observations  whose  weight  is  unity,  and  the  same  is  true 
in  the  case  of  nf,  nf/,  &c.  We  thus  resolve  the  given  values  into 
p  +  pf  +  p"  +  •  •  •  •  observations  of  the  weight  unity,  and  the  arith- 
metical mean  of  all  these  gives,  for  the  most  probable  value  of  x, 

_pn  +  p'ri  +  p"n"  +  Ac.  _  [pn\  , 

'         '      &c.          '  ' 


The  unit  of  weight  being  entirely  arbitrary,  it  is  evident  that  the 
relation  given  by  this  equation  is  correct  as  well  when  the  quantities 
P)  p'y  P">  &c-  are  fractional  as  when  they  are  whole  numbers.  The 
weight  of  XQ  as  determined  by  (41)  is  expressed  by  the  sum 


and  the  probable  error  of  x0  is  given  by 


r'  (42) 


when  r,  denotes  the  probable  error  of  an  observation  whose  weight 
is  unity.  The  value  of  r,  must  be  found  by  means  of  the  observa- 
tions themselves.  Thus,  there  will  be  p  residuals  expressed  by 
n  —  xQ,  p'  residuals  expressed  by  n'  —  o?0,  and  similarly  in  the  case  of 
n",  n"r,  &c.  Hence,  according  to  equation  (31),  we  shall  have 


r,  =  0.6745 


in  which  m  denotes  the  number  of  values  to  be  combined,  or  the 
number  of  quantities  n,  nf,  n",  &c.  For  the  mean  error  of  #0,  we 
have  the  equations 


(44; 


If  different  determinations  of  the  quantity  x  are  given,  for  which 
the  probable  errors  are  r,  r',  r",  &c.,  the  reciprocals  of  the  squares 
of  these  probable  errors  may  be  taken  as  the  weights  of  the  respective 
values  n,  n',  nrf,  &c.,  and  we  shall  have 

!L    L  ?L-    I   ^ L 

.,2  ~r  ^/2  ~r  v//2  T  •  •  *  • 


376  THEORETICAL   ASTRONOMY. 

with  the  probable  error 

=—  '  (46) 


The  mean  errors  may  be  used  in  these  equations  instead  of  the  pro- 
bable errors. 

134.  The  results  thus  obtained  for  the  case  of  the  direct  observa- 
tion of  the  quantity  sought,  are  applicable  to  the  determination  of 
the  conditions  for  finding  the  most  probable  values  of  several  un- 
known quantities  when  only  a  certain  function  of  these  quantities  is 
directly  observed.  In  the  actual  application  of  the  formulae  it  will 
always  be  possible  to  reduce  the  problem  to  the  case  in  which  the 
quantity  observed  is  a  linear  function  of  the  quantities  sought.  Thus, 
let  V  be  the  quantity  observed,  and  £,  y,  £,  &c.  the  unknown  quan- 
tities to  be  determined,  so  that  we  have 


Let  £0,  yw  f  0,  &c.  be  approximate  values  of  these  quantities  supposed 
to  be  already  known  by  means  of  previous  calculation,  and  let  x,  y, 
z,  &c.  denote,  respectively,  the  corrections  which  must  be  applied  to 
these  approximate  values  in  order  to  obtain  their  true  values.  Then, 
if  we  suppose  that  the  previous  approximation  is  so  close  that  the 
squares  and  products  of  the  several  corrections  may  be  neglected,  we 
have 

_     T,       dV     ,   dV     .   dV 

v-v°=d!x+^y+~dtz+--> 

and  thus  the  equation  is  reduced  to  a  linear  form.  Hence,  in  general, 
if  we  denote  by  n  the  difference  between  the  computed  and  the  ob- 
served value  of  the  function,  and  similarly  in  the  case  of  each  obser- 
vation employed,  the  equations  to  be  solved  are  of  the  following 
form  :  — 

ax   -f-  by   "h  cz    ~f~  du   -\-  ew    -f-  /£    -j~  n  —  0, 
a'x  +  b'y  -f  c'z  +  d'u  +  e'w  +ft  +  ri  =  0,  (47) 

a"x  4.  y'y  4-  c"z  4.  d"u  -f  e"w  +ft  +  n"  =  0, 
&c.  &c. 

which  may  be  extended  so  as  to  include  any  number  of  unknown 
quantities.  If  the  number  of  equations  is  the  same  as  the  number 
of  unknown  quantities,  the  resulting  values  of  these  will  exactly 
satisfy  the  several  equations;  but  if  the  number  of  equations  exceeds 
the  number  of  unknown  quantities,  there  will  not  be  any  system  of 


METHOD   OF   LEAST   SQUARES.  »377 

values  for  these  which  will  reduce  the  second  members  absolutely  to 
zero,  and  we  can  only  determine  the  values  for  which  the  errors  for 
the  several  equations,  which  may  be  denoted  by  v,  v',  v",  &c.,  will  be 
those  which  we  may  regard  as  belonging  to  the  most  probable  values 
of  the  unknown  quantities. 

Let  J,  Jr,  J",  &c.  be  the  actual  errors  of  the  observed  quantities; 
then  the  probability  that  these  occur  in  the  case  of  the  observations 
used  in  forming  the  equations  of  condition,  will  be  expressed  by 


and  the  most  probable  values  of  the  unknown  quantities  will  be  those 
which  make  P  a  maximum.  The  form  of  the  function  <p  (  J)  has 
been  already  found  to  be 


and  hence  we  shall  have 


m  being  the  number  of  observations  or  equations  of  condition.  In 
order  that  P  may  be  a  maximum,  the  value  of 

A2  J2  +  h'2  J'2  +  h"2  A"*  +  &c. 

must  be  a  minimum.  If  the  observations  are  equally  good,  the  ex- 
pression for  P  becomes 

p  _       hm  h2  ( A2  +  A'Z  +  A"2  +  Ac.) 

JT  —    / —  e  j 

Vitm 

and  the  condition  of  a  maximum  probability  requires  that 

J2  _j-   J'»  +   J"2  +   &c> 

shall  be  a  minimum.  Hence  it  appears  that  when  the  observations  are 
equally  precise,  the  most  probable  values  of  the  unknown  quantities 
are  those  which  render  the  sum  of  the  squares  of  the  residuals  a 
minimum,  and  that,  in  general,  if  each  error  is  multiplied  by  its 
measure  of  precision,  the  sum  of  the  squares  of  the  products  thus 
formed  must  be  a  minimum. 

If  we  denote  the  actual  residuals  by  v,  vf,  v",  &c.,  and  regard  the 
observations  as  having  the  same  measure  of  precision,  the  condition 
that  the  sum  of  their  squares  shall  be  a  minimum  gives 

dM=0          ^M  =  0>         ^W^o.&c., 

dx  *          dy  dz 


378  THEOEETICAL   ASTRONOMY. 

or 

dv    .     ,dv'         ,,dv"   , 

V  ~T-  +  v  -J-  +  *>    -T-  + —  0, 

eta    '       eta  n         da?  ~ 

dv    .     ,dvr  dv" 


I          p  **€/         .          ff  UU  .  - 

V^  +  V^  +  V    dF +----  =  °> 
&c.  &c. 

If  we  differentiate  the  equations 

ax    -{-by   +  cz   -f-  du   -j-  ew    -(-/£    +  n   =  v, 
a'z  +  6'y  -f-  c'z  -f  d'w  +  e'w  -\-ft  +  ^'  =v',  (49) 

a"a;  4.  5"y  _j_  c"z  4_  rf"M  +  e"w;  +f"t  +  n"  =  i;'r, 
&c.  &c. 

with  respect  to  x,  y,  z,  &c.,  successively,  we  obtain 

dv  _  dv'         f  dv"        „    o 

(50) 
dv       ,  dv          f  dv          lf 

dy  dy  dy 

&c.  &c.  &c. 

Introducing  these  values  into  the  equations  (48),  and  substituting  for 
0,  vr,  vff,  &c.  their  values  given  by  (49),  we  get 


[aa]  x  -j-  [a&]  y  -f  [ac]  z  -j-  [ad]  w  +  [ae]  w  +  [a/]  <  +  [a^]  =  0, 
[a6]  x  +  [65]  y  +  [6c]  2  +  [  W]  w  +  \be]  w  +  [6/]  t  +  [6/1]  =  0, 
[ac]  a;  +  [6c]  y  +  [cc]  z  -f  [cd]  u  +  [ce]  ti;  +  [c/]  *  +  [c/i]  =  0, 
\_ad~]  x  +  [6d]  y  +  [cd]  3  +  [dd]  u  +  [de]  w  +  [d/]  *  +  [dw]  =  0, 
[ae]  a;  +  {be]  y  +  [ce]  2  +  \de\  u  +  [ee]  w  -f  [e/]  <  +  [en]  =  0, 

[«/]*  -h  P/]y  +  ['/]*  +  [d/]«  +  [>/]*  +  [//]  *  +  [>]  -  0, 


in  which 

[aa]  =  aa  -f-  a'a'  -j-  a"  a"  -f-  .  .  .  . 


[ac]  =ac-f-aV+a'V'  +  .... 
[66]  =  66  -f  W  -f  V'b"  +  ____ 
&c.  &c. 

The  equations  of  condition  are  thus  reduced  to  the  same  number  as 
the  number  of  the  unknown  quantities,  and  the  solution  of  these 
will  give  the  values  for  which  the  sum  of  the  squares  of  the  residuals 
will  be  a  minimum.  These  final  equations  are  called  normal  equations. 
When  the  observations  are  not  equally  precise,  in  accordance  with 
the  condition  that  AV  +  h/2v'2  -\-  htt2vm-\-  &c.  shall  be  a  minimum, 


METHOD   OF   LEAST   SQUARES.  379 

each  equation  of  condition  must  be  multiplied  by  the  measure  of 
precision  of  the  observation;  or,  since  the  weight  is  proportional  to 
the  square  of  the  measure  of  precision,  each  equation  of  condition 
must  be  multiplied  by  the  square  root  of  the  weight  of  the  observa- 
tion, and  the  several  equations  of  condition,  being  thus  reduced  to 
the  same  unit  of  weight,  must  be  combined  as  indicated  by  the  equa- 
tions (51). 

135.  It  will  be  observed  that  the  formation  of  the  first  normal 
equation  is  eifected  by  multiplying  each  equation  of  condition  by 
the  coefficient  of  x  in  that  equation  and  then  taking  the  sum  of  all 
the  equations  thus  formed.  The  second  normal  equation  is  obtained 
in  the  same  manner  by  multiplying  by  the  coefficient  of  y;  and  thus 
by  multiplying  by  the  coefficient  of  each  of  the  unknown  quantities 
the  several  normal  equations  are  formed.  These  equations  will  gene- 
rally give,  by  elimination,  a  system  of  determinate  values  of  the 
unknown  quantities  #,  y,  z,  &c.  But  if  one  of  the  normal  equations 
may  be  derived  from  one  of  the  others  by  multiplying  it  by  a  con- 
stant, or  if  one  of  the  equations  may  be  derived  by  a  combination  of 
two  or  more  of  the  remaining  equations,  the  number  of  distinct  rela- 
tions will  be  less  than  the  number  of  unknown  quantities,  and  the 
problem  will  thus  become  indeterminate.  In  this  case  an  unknown 
quantity  may  be  expressed  in  the  form  of  a  linear  function  of  one  or 
more  of  the  other  unknown  quantities.  Thus,  if  the  number  of 
independent  equations  is  one  less  than  the  number  of  unknown 
quantities,  the  final  expressions  for  all  of  these  quantities  except  one, 
will  be  of  the  form 

X  =  a  +  Pt,  y  =  a'  -f  fit,  S  =  a"  -f  fi't,  &C.          (53^) 

The  coefficients  a,  /9,  a/,  /9',  &c.  depend  on  the  known  terms  and  co- 
efficients in  the  normal  equations,  and  if  by  any  means  t  can  be  de- 
termined independently,  the  values  of  x,  y,  z,  &c.  become  determinate. 
It  is  evident,  further,  that  when  two  of  the  normal  equations  may  be 
rendered  nearly  identical  by  the  introduction  of  a  constant  factor,  the 
problem  becomes  so  nearly  indeterminate  that  in  the  numerical  appli- 
cation the  resulting  values  of  the  unknown  quantities  will  be  very 
uncertain,  so  that  it  will  be  necessary  to  express  them  as  in  the  equa- 
tions (53). 

The  indeterrnination  in  the  case  of  the  normal  equations  results 
necessarily  from  a  similarity  in  the  original  equations  of  condition, 
and  when  the  problem  becomes  nearly  indeterminate,  the  identity  of 


380  THEORETICAL   ASTRONOMY. 

the  equations  will  be  closer  in  the  normal  equations  than  in  the  equa- 
tions of  condition  from  which  they  are  derived.  It  should  be  observed, 
also,  that  when  we  express  x}  y,  z,  &c.  in  terms  of  t,  as  in  (53),  the 
normal  equation  in  t,  which  is  the  one  formed  by  multiplying  by  the 
coefficient  of  t  in  each  of  the  equations  of  condition,  is  not  required. 

136.  The  elimination  in  the  solution  of  the  equations  (51)  is  most 
conveniently  effected  by  the  method  of  substitution.  Thus,  the  first 
of  these  equations  gives 

lab']          [ae]          [ad]          [ae]  [a/]          [an]  . 

it/        •  —  —  -  —  7/  ™~  "p       ^r  Z  '       ~p  -  ^  Zv  ~"  "*  r-       -.    IV  ~^~*  1^  ^^—  . 

[aa]  [aa]          [aa]          \_aa\  \_aa]          [aa] 

and  if  we  substitute  this  for  x  in  each  of  the  remaining  normal  equa- 
tions, and  put 


[66]  -  |g|  [at]  =  [M.1],  [fc]  -  [gj  [ao]  =  [Jc.1], 


(54) 


r    i  r    i 

[«]  -  gj  [ae]  =  [ce.ll,  [ffl  -  jgj-  [«/]  =  [c/1]  ; 


[dd]  -          [afl  =  [<M.l],  [a«]  -          [«]  =  [dV.ll 

r      T-  (56) 


=  [fai.ll  [«]  -          [^  =  [<OT-^' 

|_(ZU'J 

-C*-1!  [en]-[[«»]  =  [e».l],     (58) 


we  obtain 


METHOD   OF   LEAST  SQUARES.  381 


[66.1]  y  4  [6c.l]  z  4  \_U.\-\  u  +  [6e.l]  w  +  [6/.1]  *  +  [fcn.l]  =  0, 
[6c.l]  y  4  [cc.l]  ^  4  [cd.l]  u  4  [ce.l]  u;  4  [c/.l]  <  +  [crc.l]  =  0, 
[W.1]  y  4  [crf.l]  2  H-  [<M.l]  M  4  [efe.l]  w  +  [d/.l]  *  4  [dn.l]  =  0,     (59) 
O.I]  y  4  [ce.l]  2  4  [<I«.l]  M  +  [ec.l]  w  +  [e/.l]  <  +  [en*l]  =  0, 
+  [Cf.l]  «  +  [rf/.l]  u  +  [e/.l]  w  +  [//I]  «  +  !>•!]  -  0. 


These  equations  are  symmetrical,  and  of  the  same  form  as  the  normal 
equations,  the  coefficients  being  distinguished  by  writing  the  numeral 
1  within  the  brackets. 

The  unknown  quantity  x  is  thus  eliminated,  and  by  a  similar  pro- 
cess y  may  be  eliminated  from  the  equations  (59),  the  resulting  equa- 
tions being  rendered  symmetrical  in  form  by  the  introduction  of  the 
numeral  2  within  the  brackets.  Thus,  we  put 


.     =  [cc.2],  . 

=  [ce.2],          [»/.!]  -  [i/.l]  = 


.  . 

and  the  equations  become 


(.63) 


[cc.2]  z  +  [crf.2]  t*  +  [ce.2]  ti;  -f  [c/.2]  <  4-  [cn.2]  =  0, 

1    [cd.2]  z  4  [dd.2]  M  4  [rfe-2]  w  4-  [rff.2]  <  4  [d»-2]  =  0, 

[ce.2]  z  +  [rfc.2]  u  4  [ec.2]  to  4  [e/.2]  *  4  [cn-2]  =  0, 

[c/.2]  z  4  Hf-2]  i*  4  [e/2]  w  4-  [//2]  «  +  [>.2]  =  0. 

To  eliminate  z  from  these  equations,  we  put 


382  THEORETICAL   ASTRONOMY. 

[ee.2]  -          l  [e(,2]  =  [ee.3], 


».2]  =  [dn.3],  [en.2]  - 

W21 
[/»-2]-g|-[e«.2]  =  [/n. 

and  we  have 


[eta.3]  M  +  [de.3]  w  -f  [d/.3]  <  +  [d».3]  =  0, 

[efe.3]  ti  +  [ee.3]  w  +  [e/.3]  «  +  [en.3]  =  0,  (68) 


Again  we  put,  in  a  similar  manner, 


.3]  -  [dn.3]  =  [6n.4],    (69) 


and  the  equations  are 

+  [671.4]  =  0, 


.  .  . 

Finally,  to  eliminate  w,  we  put 

=  [/re.5])     (71) 


and  the  resulting  equation  is 

0,  (72) 


which  gives 

[/n.5] 


The  value  of  t  thus  found  enables  us  to  derive  that  of  w  by  means 
of  the  first  of  equations  (70).  The  value  of  w  being  found,  that  of 
u  will  be  obtained  from  the  first  of  equations  (68).  In  like  manner, 
the  remaining  unknown  quantities  will  be  determined  by  means  of 
the  equations  (64),  (59),  and  (51).  The  determination  of  the  unknown 
quantities  is  thus  reduced  to  the  solution  of  the  following  system  of 
equations : 


METHOD   OF   LEAST  SQUARES.  383 

&&,  +MM  +MW  +mt  +M  =0 

[aa]        r  [aa]  [aa]         '    [aa]        r  [aa] 

3+  "'f  W+  <+  =0> 


[cc.2]    [cc.2]    [cc.2]    ' 


the  coefficients  of  which  will  have  been  found  in  the  process  of  de- 
termining the  several  auxiliary  quantities.  It  will  be  observed, 
further,  that  both  in  the  normal  equations  and  in  those  which  result 
after  each  successive  elimination,  the  coefficients  which  appear  in  a 
horizontal  line,  with  the  exception  of  the  coefficient  involving  the 
absolute  terms  of  the  equations  of  condition,  are  found  also  in  the 
corresponding  vertical  line.  The  form  of  the  notation  [66.1],  [&c.l], 
&c.  may  be  symbolized  thus  : 

=  [fir.  (ft  +  1)1  (75) 


in  which  oc,  /9,  y,  denote  any  three  letters,  and  fj,  any  numeral. 

The  equations  (74)  are  derived  for  the  case  of  six  unknown  quan- 
tities, which  is  the  number  usually  to  be  determined  in  the  correction 
of  the  elements  of  the  orbit  of  a  heavenly  body;  but  there  will  be 
no  difficulty  in  extending  the  process  indicated  to  the  case  of  a  greater 
number  of  unknown  quantities,  except  that  the  number  of  auxiliaries 
symbolized  generally  by  (75)  increases  very  rapidly  when  the  number 
of  unknown  quantities  is  increased. 

137.  In  the  numerical  application  of  the  formulae,  when  so  many 
quantities  are  to  be  computed,  it  becomes  important  to  be  able  to 
check  the  accuracy  of  the  calculation  in  its  successive  stages.  First, 
then,  to  prove  the  calculation  of  the  coefficients  in  the  normal  equa- 

tions, we  put 

a+&+c+d-fe-f/=s, 

a!  -|-  j'  _j_  c'  +  d'  +  er  +/'  =  sf,  &c. 


If  we  multiply  each  of  the  sums  thus  formed  by  the  corresponding 
absolute  term  n,  and  take  the  sum  of  all  the  products,  we  have 


384  THEORETICAL   ASTRONOMY. 

[an]  +  [5n]  +  [en]  +  [an]  +  M  +  Lfii]  =  [m].  (76) 

In  a  similar  manner,  multiplying  by  each  of  the  coefficients  in  the 
original  equations  of  condition,  we  find 

[oa]  +  [aft]  +  M  +  [ad]  +  M  +  [of]  =  M, 
[a6]  +  [ftft]  +  [ftc]  +  [ftd]  +  [be]  +  [ft/]  -  [ft*], 

[ac]  -f  M  +  [cc]  +  [cd]  +  [ce]  +  [cf]  =  [«,], 

[ad]  +  [ftd]  +  [cd]  +  [dd]  -f  [de]  +  [df  ]  =  [da], 

[ae]  +  [fte]  +  M  +  [dc]  +  [ee\  +  [e/]  -  [ea], 
[«/]  +  P/1  +  [cf]  +  Hf]  +  [«/]  +  [//]  - 


Hence  it  appears  that  if  we  compute  the  sums  «,  sr,  sr/,  s'/r,  &c.,  and 
form  [as],  [6s],  [cs],  &c.  simultaneously  with  the  calculation  of  the 
coefficients  in  the  normal  equations,  the  equation  (76)  must  be  satis- 
fied when  the  absolute  terms  of  the  normal  equations  are  correct; 
and  the  equations  (77)  must  be  satisfied  when  the  coefficients  of  the 
unknown  quantities  in  the  normal  equations  are  correct. 

The  accuracy  of  the  calculation  of  the  auxiliary  quantities  sym- 
bolized by  the  equation  (75)  may  be  proved  in  a  similar  manner. 
Thus,  we  have 


which,  by  means  of  the  first  and  second  of  equations  (77),  becomes 


=  [W]  _     [a4]  +  M  _  gi  [ac] 


or 

[ba.l]  =  [ftft.1]  -f  [6c.l]  +  [ftd.l]  +  [fte.l]  +  [ft/.l]  ;  (78) 

and  similarly  we  derive  the  expressions  for  [cs.l],  [cfe.l],  &c.  It  is 
obvious,  therefore,  that  the  calculation  of  the  coefficients  in  the  equa- 
tions (59),  (64),  (68),  and  (70)  will  be  checked  as  in  the  case  of  the 
coefficients  in  the  normal  equations,  the  auxiliaries  depending  on  8 
being  determined  as  if  s,  sf,  sff,  &c.  were  the  coefficients  of  an  addi- 
tional unknown  quantity  in  the  several  equations  of  condition.  Hence 
we  must  have,  finally, 


[«*.5]  -  1>5].  (79) 

If  we  multiply  each  of  the  equations  (49)  by  its  v,  and  take  the 
sum  of  the  several  products,  we  get 

[av]  x  -f  [bv]  y  +  [cv]  *  +  [dv]  u  -j-  [ev]  w  -f-  [/v]  t  +  \vn\  =  [w], 


METHOD   OF   LEAST   SQUARES.  385 

But,  according  to  the  equations  (48)  and  (50),  we  have,  for  the  most 
probable  values  of  the  unknown  quantities, 


[av]  =  0,  |>]  =  0,  [eo]  =  0,  &c.  ; 

and  hence 

[vn~]  =  [w].  (80) 

If  we  multiply  each  of  the  equations  (49)  by  its  n,  and  take  the  sum 
of  all  the  products  thus  formed,  substituting  [vv]  for  [wi],  there  re- 

sults 

[an]  x  -f-  [bri]  y  -f-  [en]  «  +  [dn]  u  -f-  [eri]  w  -f-  \Jri\  t  ~j-  [nn]  =  [vv]. 

Substituting  in  this  the  value  of  x  given  by  the  first  normal  equa- 
tion, it  becomes 

[bn.Y]  y  +  [cn.l]  z  +  \_dn.Y]  u  +  [en.l]  w  +  [/n.l]  <  +  [nn.l]  =  [w], 
in  which 

[n«.l]  =  [*m]-^[<m].  (81) 

Substituting,  further,  for  y  its  value  given  by  the  first  of  equations 
(59),  and  continuing  the  process  as  in  the  elimination  of  the  unknown 
quantities  by  successive  substitution,  we  obtain  the  following  equa- 
tions : 

[cn.2]  z  -f  [dn.2]  u  +  [en.2]  w  -f  [/w.2]  «  +  [wi.2]  =  [w], 
[dn.3]  w  +  [e».3]  w  +  [>.3]  t  +  [nn.3]  =  [w], 

[e».4]tc  +  [>.43<-{-[nn.4]  =  M,      (82) 

|>.5]<  +  [nn.5]  =  M, 
[nn.6]  =  [vv]. 

The  expressions  for  the  auxiliaries  [ww.2],  [nn.3],  &c.  are 
[«».2]  =  [»«.!]  -  M  [in.1],  [nn.8]  =  [n».2]  -  M  [m.2], 

[nn.4]  =  [nn.8]  -  [dn.3],  [»n.5]  =  [nn.4]  -  [on.4], 


«.5].  (83) 

The  process  here  indicated  may  be  readily  extended  to  the  case  of  a 
greater  number  of  unknown  quantities,  and  we  have,  in  general,  when 
u  denotes  the  number  of  unknown  quantities, 

[vv]  =  [nn.fi"].  (84) 

25 


386  THEORETICAL   ASTRONOMY. 

This  equation  affords  a  complete  verification  of  the  entire  numerical 
calculation  involved  in  the  determination  of  the  unknown  quantities 
from  the  original  equations  of  condition.  Thus,  after  the  elimination 
has  been  completed,  we  substitute  the  resulting  values  of  x,  yy  2,  &c. 
in  the  equations  of  condition,  and  derive  the  corresponding  values 
of  the  residuals  v,  vf,  v",  &c.  Then,  taking  the  sum  of  the  squares 
of  these,  the  equation  (84)  must  be  satisfied  within  the  limits  of  the 
unavoidable  errors  of  calculation  with  the  logarithmic  tables  em- 
ployed. If  this  condition  is  satisfied,  it  may  be  inferred  that  the 
entire  calculation  of  the  values  of  the  unknown  quantities  from  the 
given  equations  of  condition  is  correct. 

138.  If  the  values  of  x,  y,  z,  &c.  thus  found  were  the  absolutely 
exact  values,  the  residuals  v,  v',  v",  &c.  would  be  the  actual  errors 
of  observation.  But  since  the  results  obtained  only  furnish  the  most 
probable  values  of  the  unknown  quantities,  the  final  residuals  may 
differ  slightly  from  the  accidental  errors  of  observation.  Further, 
it  is  evident  that  the  degree  of  precision  with  which  the  several 
unknown  quantities  may  be  determined  by  means  of  the  data  of  the 
problem  may  be  very  different,  so  that  it  is  desirable  to  be  able  to 
determine  the  relative  weights  of  the  different  results. 

It  will  be  observed  that  the  expressions  for  either  of  the  unknown 
quantities  resulting  from  the  elimination  of  the  others  is  a  linear 
•"unction  of  n,  nf,  n",  &c.,  so  that  we  have 

x  +  an  +  a!n'  +  a"n"  +  a"  V"  +  ....==  0,  (85) 

in  wLich  the  coefficients  a,  a/,  a/',  &c.  are  functions  of  the  several 
coefficients  of  the  unknown  quantities  in  the  equations  of  condition. 
If  we  now  suppose  the  equations  of  condition  to  be  reduced  to  the 
same  unit  of  weight,  the  mean  error  of  the  several  absolute  terms  of 
the  equations  will  be  the  same,  and  will  be  the  mean  error  of  an 
observation  whose  weight  is  unity.  Thus,  if  e  denotes  the  mean 
error  of  an  observation  of  the  weight  unity,  the  mean  error  of  an 
will  be  ae,  that  of  a'n'  will  be  o/e',  and  similarly  for  the  other  terms 
of  (85, ;  and,  according  to  the  equation  (35),  the  mean  error  of  x 
will  bt) 

zx  =  £  ]/a2  -f-  a'2  -f  a"2  -f-  &C.  =  £  l/[aa].  (86) 

Hence  the  weight  of  x  will  be  expressed  by 

' 


METHOD   OF    LEAST   SQUARES.  387 

Let  x,  denote  the  true  value  of  x,  namely,  that  which  would  be 
obtained  if  the  true  values  of  v,  vf,  v",  &c.  were  retained  in  the 
second  members  of  the  equations  of  condition  instead  of  putting 
them  equal  to  zero  ;  then  it  is  evident  that  the  expression  for  x,  must 
be  that  which  would  result  by  substituting  n  —  v  in  place  of  n  in  the 
formulae  for  the  most  probable  value  as  determined  from  the  actual 
data.  Hence  we  have 

Xf  +  n(n  —  v-)  +  a>'-t/)  +  .  .  .  .  =  0, 
and  comparing  this  with  the  expression  (85),  we  obtain 


Substituting  in  this  the  values  of  v,  vf,  v",  &c.  given  by  the  equations 
(49),  there  results 

x,  =  x+  [aa]  X,  -f  [06]  y,  +  [ac]  z,  +  [ad]  u,  -f  [oe]  w,  -f  [a/]  t,  +  [aw], 

and  since,  according  to  (85),  x  -j-  [an]  =  0,  in  order  to  satisfy  this 
expression  for  xn  we  must  evidently  have 

[aa]  =  l,      [06]  =0,      [ac]=0,      [ad]  =  0,      [ae]  =  0,      [a/]  =  0.  (88) 

Since  the  values  of  the  unknown  quantities  as  determined  by  the 
normal  equations  must  be  the  same  by  whatever  mode  the  elimination 
may  have  been  performed,  let  us  suppose  the  method  of  indeterminate 
multipliers  to  be  applied  for  the  determination  of  a?,  and  let  these 
multipliers  be  designated  by  9,  qf,  q",  &c.  ;  then,  the  values  of  these 
factors  are  determined  by  the  condition  that  the  coefficient  of  x  in 
the  final  equation  shall  be  unity,  and  that  the  coefficients  of  the  other 
unknown  quantities  shall  be  zero.  Hence  we  shall  have 

[aa]  q  +  [oft]  qf  +  [ac]  g"  +  [ad]  q'"  +  ....  =  1, 
lab]  q  +  [56]  ^  +  [be]  q"  +  [bd]  q'"  +....  =  0,  (89) 

[ac]  q  +  [be]  q'  +  M  5"  +  M  5"'  -f  .  .  .  .  =  0, 
&c.  &c. 

and  also,  retaining  the  residuals  v,  vf,  v",  &c.  in  the  formation  of  the 
normal  equations, 


Therefore,  since 

Xf  +  [an]  =  [av], 

and  since  the  first  member  of  this  equation  must  be  identical  with 
the  first  member  of  (90),  we  have 


388  THEORETICAL   ASTRONOMY. 

which  gives,  by  expanding  the  several  sums, 

aq    +  bq'    +  cq"    +  dqm    +....  =  «, 
a'q  -f  Vq'  +  e'q"  +  d'q'"  +  ....  =  •',  (91) 

a"?  +  &'Y  +  *'Y'  +  <*'Y"  4-  •  •  •  •  =  a", 
&c.  &c. 

Multiplying  each  of  these  equations  by  its  cc,  and  adding  the  pro- 
ducts, the  result  is 

[«a]  q  +  [06]  qf  +  [«c]  3"  +  [ad]  9'"  +  ....==  [aa], 
which,  by  means  of  the  equations  (88),  reduces  to 

9  4-  (92) 

Hence  it  appears  that  the  eliminating  factor  q  is  the  reciprocal  of  the 
weight  of  x,  and,  since  the  coefficients  of  q,  qr,  qff,  &c.  in  the  equa- 
tions (89)  are  the  same  as  those  of  x,  y,  z,  &c.  in  the  normal  equa- 
tions, that  if  we  put  [an]  =  —  1,  \bn\  =  0,  [en]  =  0,  &c.,  in  the 
normal  equations,  the  resulting  value  of  x  will  be  the  reciprocal  of 
the  weight  of  the  most  probable  value  of  this  quantity. 

The  equation  (90)  shows  that  if,  in  the  general  elimination,  by 
whatever  method  it  may  have  been  effected,  we  write  [av],  [bv],  &c. 
instead  of  zero  in  the  second  members  of  the  normal  equations  re- 
spectively, the  coefficient  of  [at?]  is  the  reciprocal  of  the  weight  of  x. 
It  is  obvious  that  it  will  not  be  necessary  to  know  the  numerical 
values  of  [av\9  [bv],  &c.,  since  only  the  coefficient  q  is  required.  The 
most  probable  value  of  x  is  found  from  (90)  by  the  condition  of  a 
minimum  of  the  squares  of  the  residuals,  namely,  that 

[av]  =  0,         [bv]  =  0,         [co]  =  0,         &c. 

The  process  here  indicated  for  the  determination  of  the  weight  of 
the  final  value  of  x  is  general,  and  applies  to  the  case  of  any  other 
unknown  quantity  provided  that  the  necessary  changes  are  made  in 
the  notation.  Thus,  the  reciprocal  of  the  weight  of  y  is  determined 
by  writing,  in  the  normal  equations,  —  1  in  place  of  [bri],  and  putting 
[an],  [en],  &c.  equal  to  zero,  and  completing  the  elimination.  It 
is  also  the  coefficient  of  [bv]  in  the  value  of  y  when  the  elimination 
is  effected  with  the  symbols  [av],  [bv],  &c.  retained  in  the  second 
members  of  the  normal  equations. 

139.  It  may  be  easily  shown  that  when  the  elimination  is  effected 
by  the  method  of  successive  substitution,  as  already  explained,  the 


METHOD   OF   LEAST  SQUARES.  3<S9 

coefficient  of  the  unknown  quantity  which  is  made  the  last  in  the 
elimination,  in  the  final  equation  for  its  determination,  is  equal  to  the 
weight  of  the  resulting  value  of  that  quantity.  Thus,  in  the  case  of 
the  equations  for  six  unknown  quantities,  since  the  reciprocal  of  the 
weight  of  the  most  probable  value  of  t  is  the  value  of  t  obtained 
from  the  normal  equations  by  putting  [/n]  =  —  i,  and  [an],  [bn],  [en], 
&c.  equal  to  zero,  the  equations  (63),  (67),  (69),  and  (71)  show  that 
we  have 

[>]  =  I>-1]  =  l>-2]  =  |>.3]  =  |>.4]  =  [/n.5]  =  —  1, 
and  hence,  according  to  (72),  for  the  reciprocal  of  the  weight  of  t, 


which  gives 

(93) 


The  weight  of  t  is  therefore  equal  to  its  coefficient  in  the  final  equa- 
tion which  results  from  the  elimination  of  the  other  unknown  quan- 
tities by  successive  substitution.  Hence,  by  repeating  the  elimination, 
successively  changing  the  order  of  the  quantities,  so  that  each  of  the 
unknown  quantities  may  have  the  last  place,  the  weights  will  be 
determined  independently,  and  the  agreement  of  the  several  sets  of 
values  for  the  unknown  quantities  will  be  a  proof  of  the  accuracy  of 
the  calculation.  It  is  not  necessary,  however,  to  make  so  many 
repetitions  of  the  elimination,  since,  in  each  case,  the  weights  of  two 
of  the  unknown  quantities  will  be  given  by  means  of  the  auxiliaries 
used  in  the  elimination.  Thus,  the  reciprocal  of  the  weight  of  w  is 
obtained  by  putting  [eri\  =  —  1,  and  the  other  absolute  terms  of  the 
normal  equations  equal  to  zero,  and  finding  the  corresponding  value 
of  w.  This  operation  gives 


Hence  the  equation  (73)  becomes 

t  =  —  •=- 


and  substituting  this  value  of  t  in  the  last  of  equations  (70),  we  get 

f.41 

' 


or 

<*.4],  (94) 


390  THEOEETICAL   ASTKONOMY. 

which  gives  the  weight  of  w  in  terms  of  the  auxiliary  quantities 
required  in  the  determination  of  its  most  probable  value. 

If  the  order  of  elimination  is  now  completely  reversed,  so  that  x 
is  made  the  last  in  the  elimination,  the  weights  of  x  and  y  will  be 
determined  by  the  equations 

p.==[oa.5], 

^=M[M.4]. 

•*»       [aa.4]  L 

A  third  elimination,  in  which  z  and  u  are  the  unknown  quantities 
first  determined,  will  give  the  weights  of  these  determinations.  It 
appears,  therefore,  that  when  only  four  unknown  quantities  are  to  be 
found,  a  single  repetition  of  the  elimination,  the  order  of  the  quan- 
tities being  completely  reversed,  will  furnish  at  once  the  weights  of 
the  several  results,  and  check  the  accuracy  of  the  calculation.  When 
there  are  only  two  unknown  quantities,  the  elimination  gives  directly 
the  values  of  these  quantities  and  also  of  their  weights. 

140.  In  the  case  of  three  or  more  unknown  quantities,  the  weights 
of  all  the  results  may  be  determined  without  repeating  the  elimina- 
tion when  certain  additional  auxiliary  quantities  have  been  found. 
The  weights  of  the  two  which  are  first  determined  are  given  in  terms 
of  the  auxiliaries  required  in  the  elimination,  that  of  the  quantity 
which  is  next  found  will  require  the  value  of  an  additional  auxiliary 
quantity,  the  succeeding  one  will  require  two  additional  auxiliaries, 
and  so  on.  The  equations  (74)  show  that  when  the  substitution  is 
effected  analytically  the  final  value  of  x  will  have  the  denominator 

D  =  [ad]  [66.1]  [cc.2]  [eW.3]  [ee.4]  [jgf.5], 

and  this  denominator,  being  the  determinant  formed  from  all  the 
coefficients  in  the  normal  equations,  must  evidently  have  the  same 
value  whatever  may  be  the  order  in  which  the  unknown  quantities 
are  eliminated.  Let  us  now  suppose  that  each  of  the  unknown 
quantities  is,  in  succession,  made  the  last  in  the  elimination,  and  let 
the  auxiliaries  in  each  elimination  be  distinguished  from  those  when 
t  is  last  eliminated  by  annexing  the  letter  which  is  the  coeffi  3ient  rf 
the  quantity  first  determined ;  then  we  shall  have 

D  =  [ad]  [66.1]  [cc.2]  [dd.3]  \_eeA~]  [ff.5] 
=  M,  [W.H  [<*.2].  [<W.3].  UfAl  0*5] 
=  [_aa\  [66.1],  [cc.2]d  0*3],  [jfjf.4],  [cW.5] 
=  [oa],  [66.1]c  [eta.2],  [ee.3]c  [//.4]c  [cc.5] 

5  [66.5] 


METHOD   OF   LEAST  SQUARES.  391 

It  will  be  observed,  however,  that  when  the  order  of  elimination  is 
changed,  only  those  auxiliaries  which  involve  the  coefficient  of  the 
quantity  which  is  made  the  last  in  the  changed  order  will  be  changed. 
Hence,  if  we  add  the  distinguishing  letter  only  to  those  auxiliaries 
•yhich  have  a  different  value  in  the  new  order,  we  have 


D  =  [aa]  [66.1]  [cc.2]     [oU3]  [ee.4] 

=  [aa]  [66.1]  [cc.2]     [dd.3]  [//.4]  [ee.5] 
=  [aa]  [66.1]  [cc.2]     [66.3]   [//.4]d  [dd.5] 
=  [aa]  [66.1]  [dW.2]    [66.3].  [//.4]c  [cc.5] 
=  [aa]  [cc.l]   [aU2]6  [66.3]6  [//.4]6  [66.5] 
-  [66]  [cc.1].  [cW.2].  [66.3].  [//.4]a  [aa.5], 


and  from  these  equations  we  obtain 


[//•5] 
"  [jfif.4] 


p   == 

I       M     -+    I  1    /^    •  "*    I 

(96) 

p  =|_cc.5j    r=^.'-'^1        IOC-'±I       '^-01   -     -_ 


[66.4] 

[<W.3], 

-[cc.2], 
[cc.2] 

-  [66.1], 
[66.1] 

[66.3] 

[ee.4] 

[66.4]° 
[66.31 
[66.4] 

[oU3] 

[oU3/ 

[cc.l] 
[cc.2] 

[ee.3]. 

[cc.l]a 

[66] 

[//-4]5 

^     \aa  51  Li/- 5]  L""  -J  L-— J  tw-"J  LW-^J     r        -i 

P*  —  Laa-^J  —  r  /v/ti        r^^  QI        r^^  on       r™  n          r^/n     LaaJ> 


by  means  of  which  the  weights  of  the  six  unknown  quantities  may 
be  determined.  The  process  here  indicated  may  be  readily  extended 
to  the  case  of  a  greater  number  of  unknown  quantities.  The  equa- 
tion for  pw  is  identical  with  (94),  the  expression  for  pu  introduces  the 
new  auxiliary  quantity  [,^.4]^,  and  that  for  ps  introduces  two  new 
auxiliaries. 

The  expressions  for  the  new  auxiliaries  \JfA~\d^  [jf/".4]c,  [ee.3]c,  &c. 
are  easily  formed  by  observing  that  all  the  auxiliaries  as  far  as  those 
which  are  designated  by  the  numeral  4  are  not  affected  by  putting  e 
or /last,  that,  as  far  as  those  which  contain  the  numeral  3,  it  makes 
no  difference  whether  d,  e,  or  /  is  placed  last,  that  those  distinguished 
by  the  numerals  1  and  2  are  not  affected  by  making  c,  c/,  e,  or /the 
last,  and  that  those  designated  by  the  numeral  1  are  unchanged 
unless  a  is  made  the  last.  Thus,  we  obtain 


=  [^-p]^3]'  (97) 


•J92  THEORETICAL   ASTRONOMY. 

and,  also, 

Egl 

=  [ee.2]  - 


In  like  manner  we  may  derive  the  expressions  for  the  new  auxiliaries 
introduced  into  the  equations  for  py  and  px.  It  will  be  expedient, 
however,  in  the  actual  application  of  the  formulae,  to  eliminate  first 
in  the  order  a?,  y,  z,  u,  w,  t,  and  the  weights  of  the  results  for  u,  w, 
and  t  will  be  obtained  by  means  of  the  first  three  of  equations  (96), 
the  single  additional  auxiliary  required  being  found  by  means  of 
(97).  Then  the  elimination  should  be  performed  in  the  order  t,  w,  u, 
Zj  y,  x,  and  we  shall  have 


(99) 


by  means  of  which  the  weights  of  #,  y,  and  2  will  be  determined. 
The  agreement  of  the  two  sets  of  values  of  the  unknown  quantities 
will  prove  the  accuracy  of  the  numerical  calculation  in  the  process 
of  elimination. 

141.  The  weights  of  the  most  probable  values  of  the  unknown 
quantities  may  also  be  computed  separately  when  certain  auxiliary 
factors  have  been  found,  and  these  factors  are  those  which  are  intro- 
duced when  the  equations  (74)  are  solved  by  the  method  of  inde- 
terminate multipliers  instead  of  by  successive  substitution.  Thus, 
in  order  to  find  x,  let  the  first  of  these  equations  be  multiplied  by  1, 
the  second  by  A',  the  third  by  An  ',  the  fourth  by  A'",  and  so  on, 
and  let  the  sum  of  all  these  products  be  taken  ;  then  the  equations 
of  condition  for  the  determination  of  the  several  eliminating  factors 
will  be 


0 


(ioo) 


0  _  ,  •          A>    ,  -          An    ,  -          „„  ,      jiv 

-  [aa]  +  t  bb.l  \  A    f  [cc.2]  *     *"  ^  1  ' 

0  _  Ca/1    ,   [V-l]    .,   ,   [#2]    ,, 

U  ""  [aa]  +  IWj       +  [cc.2]  ^   ^     [dd.3]  [ee.4] 


METHOD   OF   LEAST   SQUARES.  393 

To  determine  y  from  the  last  five  of  equations  (74),  let  the  eliminating 
factors  be  denoted  by  B  ',  £"',  B*9  and  .£v,  and  we  shall  have 

[6e.l]        R,, 

" 


_       .  [eeJ.2]     „          „ 

-       +       ^      ?> 


[C6.2]      ,      [«fo.3]  ™,       „, 

f  *  ' 


«,   ,  ™      [6/4]     Iv 

fj6      fjE 


In  a  similar  manner,  we  obtain  the  following  equations  for  the  de- 
termination of  the  eliminating  factors  necessary  for  finding  the  values 
of  the  remaining  unknown  quantities  : 


[ce.2]        [rfe.3]   ~,,, 

" 


[6/4]     1T 
°      " 


(102) 


[«.4J 

The  expressions  for  the  values  of  the  unknown  quantities  will  there- 
fore become 


_ 
- 


[an]       [6ro.l]   ,,      [ot.2]  ,„     [dro.3]  .w      [en.4]  ., 
"1  "  ""^ 


[66.1]    '   [cc.2] 

r  7    en       r     AT  rf\~\  (1^3) 
\_dn.6}       \enA\  ^.  v       L/^AI  ^r 


LW1 


394  THEOKETICAL   ASTRONOMY. 

The  tii»t  of  these  equations  will  give  the  reciprocal  of  the  weight  of 
x,  when  we  put  [an]  =  —  1,  and  the  other  absolute  terms  of  the 
normal  equations  equal  to  zero;  the  second  will  give  the  reciprocal 
of  the  weight  of  y  by  putting  [bn]  =  -  1,  and  the  other  absolute 
terms  of  the  normal  equations  equal  to  zero ;  and,  continuing  the 
process,  finally  the  last  equation  will  give  the  reciprocal  of  the  weight 
of  t  when  we  put  fn  =  —  1,  and  [an],  [bn],  [en],  &c.  equal  to  zero. 
It  remains,  therefore,  to  determine  the  particular  values  of  [6n.l], 
[en. 2],  &c.,  and  the  expressions  for  the  weights  will  be  complete. 
If  we  multiply  the  first  of  equations  (100)  by  [an],  it  becomes 

\bn.l-]  =  [an-]  A'  +  [bn].  104) 

Multiplying  the  second  of  equations  (100)  by  [an],  and  the  first  of 
(101)  by  [bn],  adding  the  products,  and  introducing  the  value  of 
[6n.l]  just  found,  we  get 

[en-]  -  [cn.l]  +  j^l  [bn.l]  +  [an]  A"  +  [bn]  B"  =  0, 

which  reduces  to 

[an~]  A"  +  [bn-]  B"  -f  [en]  =  [cn.2].  (105) 

Multiplying  the  third  of  equations  (100)  by  [an],  the  second  of  (101) 
by  [bn],  and  the  first  of  (102)  by  [en],  adding  the  products,  and  re- 
ducing by  means  of  (104)  and  (105),  we  obtain 

0  =  Idn]  -  [<fa.l]  +  ^  [Jn.1] 

+  gj  [e».2]  +  [on]  A"1  +  [in]  £'"  +  [«•»]  C"", 

which,  by  means  of  the  expressions  for  the  auxiliaries,  is  further  re- 
duced to 

[an-]  A"  +  [bn]  B'"  +  [en]  C'"  +  [dn]  =  [dn.3].  (106) 

In  a  similar  manner  we  find,  from  the  remaining  equations  of  (100), 
(101),  and  (102),  the  following  expressions: 

[an]  A*  +  [bn^  B»+  [cm]  C*  +  \dn\  1F+  [>]  =  [en.4], 

[an]  A*  +  [bn]  Bv  +  [en]  <7V  +  \dn\  Dv  -f  [en]  E*  +  [>]  =  [/7i.5].  ^ 

The  equations  (104),  (105),  (106),  and  (107),  enable  us  to  find  the 
particular  values  of  [6n.l],  [cn.2],  &c.  required  in  the  expressions  for 
the  reciprocals  of  the  weights.  Thus,  for  the  weight  of  x,  we  have 

Ian]  =  —  1,          [bn]  =  [m]  =  \dn\  =  [en]  =  [fn]  =  0 ; 


METHOD   OF   LEAST  SQUARES.  395 

and  these  equations  give 

[^.1]  =  _  A't  [en.2]  =  —  A",  [dn.3]  =  -  A'", 

[6W.4]  =  —  Aiv,  L/w.5]  =  —  A\ 

For  the  case  of  the  weight  of  y,  we  have 

[bn]  =  —  1,  [cm]  =  [CTI]  =  [drc]  =  [en]  =  [/w]  =  0, 

and  the  same  equations  give 

p?l.l]  =  —  1,  [cn.2]  =  —  J3",  [dn.3]  =  —  J3'", 

[en.4]  =  —  Biv,  L/k5]  =  —  £v. 

We  have,  also,  for  the  weight  of  z, 

|_cn.2]  =  — 1,         [cfri.3]=  —  C"",         [en.4]  =  —  Civ,         |>.5]  =  — C\ 

for  the  weight  of  w, 

[dn.3]  =  —  1,  [m.4]  =  —  Dly,  [./h.5]  =  —  7)T ; 

for  the  weight  of  w, 

[«i.4]  =  -l,  [/n.5]  =  -^; 

and  finally,  for  the  weight  of  £, 

[/«.5]=    -1. 

Introducing  these  particular  values  into  the  equations  (103),  the  cor- 
responding values  of  the  unknown  quantities  are  the  reciprocals  of 
the  weights  of  their  most  probable  values,  respectively;  and  hence 
we  derive 

1  AA'        A"A'        A'"A>"         ^1Vj[iV        AVA* 


]7  ~  [oa]  ^  [56.1]  "*"  [cc.2]  n"  [dd.3]  T  [ee.4]  T  [jfif.5]  ' 
1       _J  __    jy^y       B'^^^       B*B* 
y  ~  [SO]  +  [^2j  ""  [dd.3]  +  [e6.4]  " 
1  1          C'"  C'"      CIVCIV       CVCV 

[SSI  H'  LPT 


1  _        1  EVEV 

~Wl 

1        1 


The  equations  (103)  and  (108)  will  serve  to  determine  separately 
the  value  of  each  unknown  quantity  and  also  that  of  its  weight,  the 


396  THEORETICAL   ASTRONOMY. 

auxiliary  factors  A'  ',  A",  B",  &c.  having  been  found  from  the  equa- 
tions (100),  (101),  and  (102).  If  we  reverse  the  operation  and  re- 
compose  the  equations  (74)  by  means  of  the  expressions  for  the  un- 
known quantities  given  by  (103),  the  conditions  which  immediately 
follow  furnish  another  series  of  equations  for  the  determination  of  the 
auxiliary  factors.  The  ea  nations  thus  derived  will  give  first  the  values 
of  A',  B",  <?'",  Div,  and  Ev  ;  then,  those  of  A",  B'",  Civ,  Z>v;  and  so 
on.  They  are  equally  as  convenient  as  those  already  given,  provided 
that  the  values  of  all  the  unknown  quantities  are  required  as  well  as 
their  respective  weights. 

142.  The  formulae  already  given  for  the  relations  between  the  data 
of  the  problem  and  the  weights  of  the  most  probable  values  of  the 
unknown  quantities,  are  those  which  are  of  the  greatest  practical 
value.  It  will  be  apparent  from  what  has  been  derived  that  there 
must  be  a  variety  of  methods  which  may  be  applied,  but  that  all  of 
these  methods  involve  essentially  the  same  numerical  operations. 
The  peculiar  symmetry  of  the  normal  equations  affords  also  a  variety 
of  expressions  applicable  to  the  different  phases  under  which  the 
problem  presents  itself. 

According  to  the  general  theory  of  elimination,  the  expression  for 
any  unknown  quantity,  as  determined  from  the  normal  equations, 
may  be  put  in  the  form 

*=    -|[«»]-f  [i»]-^M-&c,  (109) 

in  which  D  is  the  determinant  formed  from  all  the  coefficients  of  tho 
unknown  quantities  in  the  normal  equations,  and  in  which  A,  A',  A", 
&c.  are  the  partial  determinants  required  in  the  elimination.  Thus, 
A  is  the  determinant  formed  from  the  coefficients  of  all  the  unknown 
quantities  except  x,  in  all  the  equations  except  the  first;  A"  is  the 
determinant  formed  from  the  coefficients  of  y,  z,  &c.  in  all  the  equa- 
tions except  the  second  ;  and  the  values  of  An  r,  Af",  &c.  are  formed 
in  a  similar  manner.  Now,  since  the  value  of  x  which  results  when 
we  put  [an]  —  —  1,  and  the  other  absolute  terms  of  the  normal 
equations  equal  to  zero,  is  the  reciprocal  of  the  weight  of  the  most 
probable  value  of  this  unknown  quantity  as  given  by  (109),  we  have 


In  like  manner,  the  expression  for  the  most  probable  value  of  y  will  be 


METHOD    OF   LEAST   SQUARES,  397 

y=   -M-     [^]-M-&c.,  (ill) 


B,  B',  B",  &c.  being  the  partial  determinants  formed  when  the  co- 
efficients of  y  are  omitted;  and  for  its  weight  we  have 


The  formulae  for  the  most  probable  value  of  z  and  for  its  weight  are 
entirely  analogous  to  those  for  x  and  yy  so  that  the  process  here  indi- 
cated may  be  extended  to  the  case  of  any  number  of  unknown  quan- 
tities. It  appears,  therefore,  that  the  weight  of  the  most  probable 
value  of  any  unknown  quantity  is  found  by  dividing  the  complete 
determinant  of  all  the  coefficients  by  the  partial  determinant  formed 
when  we  omit  the  normal  equation  corresponding  particularly  to  this 
unknown  quantity,  and  when  we  omit  also  the  coefficients  of  this 
quantity  in  the  remaining  normal  equations. 

The  peculiar  arrangement  of  the  coefficients  in  the  normal  equa- 
tions abbreviates  somewhat  the  expressions  for  the  several  determi- 
nants. Thus,  in  the  case  of  three  unknown  quantities,  we  have 

A  =  [66]  [cc]  —  [6c]2,  Bf  =  [ad]  [cc]  —  [ac]2,  C"  =  [aa]  [66]  —  [a6]2, 
D  =  [aa]  [66]  [cc]  +  2[a6]  [be]  [ac]  —  [ad]  [6c]2—  [66]  [ac]2—  [cc]  [a6]2, 

which  are  all  the  quantities  required  for  finding  simply  the  weights 
of  the  most  probable  values  of  a?,  y,  and  z.  The  expression  for  the 
weight  of  z  is 

D 


When  there  are  but  two  unknown  quantities,  we  have 

A  =  [66],  B'  =  [ad],  D  =  [aa]  [66]  —  [a6]2, 

and  hence 

_  [aa]  [66]  —  [a6]2  _  [oa]  [66]  —  [a6]2 

Px~  [66]  P*~  [aa] 

When  the  number  of  unknown  quantities  is  increased,  the  expressions 
for  the  determinants  necessarily  become  much  more  complicated,  and 
hence  the  convenience  of  other  auxiliary  quantities  is  manifest. 

143.  The  case  has  been  already  alluded  to  in  which  the  determina- 
tion of  the  values  of  the  unknown  quantities  is  rendered  uncertain 
by  the  similarity  of  the  signs  and  coefficients  in  the  normal  equations, 


398  THEORETICAL   ASTRONOMY. 

and  in  which  the  problem  becomes  nearly  indeterminate.  Sometimes 
it  will  be  possible  to  overcome  the  difficulty  thus  encountered  by  a 
suitable  change  of  the  elements  to  be  determined  ;  but,  generally,  for 
a  complete  and  satisfactory  solution,  additional  data  will  be  required. 
It  often  happens,  however,  that  several  of  the  unknown  quantities 
may  be  accurately  determined  from  the  given  equations  when  the 
values  of  the  others  are  known,  but  that  the  certainty  of  the  deter- 
mination of  the  same  quantities  is  very  greatly  impaired  when  all 
the  unknown  quantities  are  derived  simultaneously  from  the  same 
equations.  Let  us  suppose  that  one  of  the  unknown  quantities  is, 
from  the  very  nature  of  the  problem,  not  susceptible  of  an  accurate 
determination  from  the  data  employed.  The  equations  will  then 
present  themselves  in  a  form  approaching  that  in  which  the  number 
of  independent  relations  is  one  less  than  the  number  of  unknown 
quantities,  so  that  it  will  be  necessary  to  determine  the  other  unknown 
quantities  in  terms  of  that  whose  value  is  necessarily  uncertain.  In 
this  case  the  elimination  should  be  so  arranged  that  the  quantity 
which  is  regarded  as  uncertain  is  that  whose  value  would  be  first 
determined.  Then,  if  its  coefficient  in  the  final  equation,  corre- 
sponding to  (72),  is  very  small,  a  circumstance  which  indicates  at 
once  the  existence  of  the  uncertainty  when  it  is  not  otherwise  sus- 
pected, the  process  of  elimination  should  not  be  completed,  and  the 
auxiliary  quantities  should  be  determined  only  as  far  as  those  re- 
quired in  the  formation  of  the  equation  which  corresponds  to  the  first 
of  (70).  Thus,  let  t  be  the  uncertain  quantity,  and  we  have 

[6/4]          len.4] 

W  =   -   -  -  —  t  -   p  -  -pr-, 

[ee.4]          [ee.4] 

which  must  be  substituted  for  w  in  the  first  of  equations  (68).  AVe 
thus  obtain  w,  u,  z,  y,  and  x  as  functions  of  t.  If  the  solution  is 
effected  by  means  of  the  equations  (103),  let  xw  yw  ZQ,  &c.  denote  the 
values  of  these  unknown  quantities  when  we  put  £  =  0;  and  then 
we  shall  have 

_  [an]  _  [bnA.~\  [c?i.2]   ,„       [cfoi.3]   .,„       [en  A]   ,lv 

[cm]       [55.1]  [ce.2]  [cW.3]  [ee.4] 

[fokl]       [e*.2]  ™       [d»-3]  »,„       [en.4]  «v  riij  , 

2/0  ~    ~  \m\~teS\*  ~\ddX\"         [66.4]*' 

l>*.2]       [<foi.3]  _ 
' 


[cc.2]       [<W.3]  [ee.5] 


and  hence 

a:  = 


METHOD    OF   LEAST   SQUARES.  399 

[Cfa.8]         |>.4] 

[<W.8]      [«.4]  •"' 


=  w+E't.  <114J 


As  soon  as  t  is  determined  by  some  independent  condition  or  relation, 
these  equations  will  give  the  corresponding  values  of  x9  y,  z,  &c.  The 
mean  errors  of  xw  y0,  z0,  &c.  having  been  determined  by  neglecting  t 
entirely,  if  we  denote  the  mean  error  of  the  final  adopted  value  of  t 
by  e0  the  mean  errors  of  the  corresponding  values  of  the  other 
variables  will  be  given  by 

(O'+^MV,^=^+     R^J*=^E+,C^>  (115) 

in  which  (ex),  (ey),  &c.  denote  the  mean  errors  of  xw  yw  &c.  These 
formulae  show,  also,  that  when  one  of  the  variables  is  neglected,  the 
equations  assign  too  great  a  degree  of  precision  to  the  results  thus 
obtained. 

When  there  are  two  or  more  unknown  quantities  which  cannot  be 
determined  from  the  data  with  sufficient  certainty,  the  problem  must 
be  treated  in  a  manner  entirely  analogous  to  that  here  indicated;  but, 
since  cases  of  this  kind  will  rarely,  if  ever,  occur,  it  is  not  necessary 
to  pursue  the  subject  further. 

144.  The  weights  which  are  obtained  for  the  most  probable  values 
of  the  unknown  quantities  enable  us  to  find  the  mean  and  probable 
errors  of  these  values.  Let  s  denote  the  mean  error  of  an  observa- 
tion whose  weight  is  unity;  then  the  mean  error  of  x  will  be 

(116) 


and,  in  like  manner,  the  expressions  "for  the  mean  errors  of  y,  z,  u, 
&c.  will  be 

e="  '.  =  -?-  «.  =  -£=,&c.  (117) 

1/P,  Vp.  Vp, 

It  remains,  therefore,  to  determine  the  value  of  e  by  means  of  the 
final  residuals  obtained  by  comparing  the  observed  values  of  the 
function  with  those  given  by  the  most  probable  values  of  the  va- 


400  THEORETICAL   ASTRONOMY. 

riables.     If  these  residuals  were  the  actual  fortuitous  errors  of  obser- 
vation, the  mean  error  of  an  observation  would  be 

M 


m  being  the  number  of  equations  of  condition.  This  value  is  evi- 
dently an  approximation  to  the  correct  result;  but  since  by  supposing 
the  residuals  v,  vf,  v",  &c.  to  be  the  actual  errors  of  the  several  ob- 
served values  of  the  function,  we  assign  too  high  a  degree  of  pre- 
cision to  the  several  results,  the  true  value  of  e  must  necessarily  be 
greater  than  that  given  by  this  equation.  Let  the  true  values  of  the 
unknown  quantities  be  x  -f-  A#,  y  -f-  Ay,  z  -f-  AZ,  &c.,  the  substitution 
of  which  in  the  several  equations  of  condition  would  give  the 
residuals  J,  Jr,  J",  &c.  ;  then  we  shall  have 

&A?/  -f-  cAz  -f-  d&u  ----  -f  v  =  A, 

'         ' 


&c.  &c. 

If  we  multiply  each  of  these  equations  by  its  J,  and  take  the  sura 
of  all  the  products,  we  get 

[o  J]  *x  -f  [6  J]  Ay  -f  [cJ]  A^  +  [dJ]  AW  +  ....  -f  [vJ]  =  [J  J]. 

But  if  we  multiply  each  of  the  same  equations  by  its  v,  take  the  sum 
of  the  products,  and  reduce  by  means  of  (48)  and  (50),  we  obtain 


_  =  [>*]; 

and  hence  we  derive 

[J  J]  =  [VV]  4  [O J]  AX  -f  [6 J]  Ay  +  [C J]  A3  +  [d J]  AW  +  ....    (119) 

If  we  form  the  normal  equations  from  (118),  it  will  be  observed  that 
they  are  of  the  same  form  as  the  normal  equations  formed  from  the 
original  equations  of  condition,  provided  that  we  write  —  A  in  place 
of  n;  and  hence,  according  to  (85),  we  have 

A*  =  aJ  +  a'J'  +  a"J''  +  ..... 

We  have,  also, 

[o J]  =  aA  +  a' A'  4  a" A"  4 , 

and  the  product  of  these  equations  gives 

[aJ]  AX  —  aa  J2  4-  aVJ'2  4  a"a" J"2  +  /. . . 
4-aa'JJ'-f  aa"JJ"4-.... 

The  mean  value  of  the  terms  containing  JJ',  Jd",.&c.  is  zero,  and 


COMBINATION   OF    OBSERVATIONS.  40J 

for  the  mean  values  of  J2,  J'2,  J"2,  &c.  we  must,  in  each  case,  write 
e2.     Hence  the  mean  value  of  the  product  [a  J]  A#  will  be 


and  this,  by  means  of  the  first  of  equations  (88),  is  further  reduced  to 

[a  J]  &x  —  £2. 

In  a  similar  manner,  we  obtain  the  value  e2  for  the  mean  value  of 
each  of  the  products  [ftzfJA?/,  [cJJA2,  &c.  Now,  the  terms  added  to 
[vv]  in  the  second  member  of  the  equation  (119)  are  necessarily  very 
small,  and,  although  their  exact  value  cannot  be  determined,  we  may 
without  sensible  error  adopt  the  mean  values  of  the  several  terms  as 
here  determined,  so  that  the  equation  becomes 

[  J  J]  =  [tw]  +  pe*,  (120) 

u  being  the  number  of  unknown  quantities.  Therefore,  since 
[  J  J]  =  me2,  we  shall  have 


m  —  fj.        *  m  —  /j. 

by  means  of  which  the  mean  error  of  an  observation  whose  weight 
is  unity  may  be  determined.  When  p  =  1,  this  equation  becomes 
identical  with  (30). 

For  the  determination  of  the  probable  errors  of  the  final  values  of 
the  unknown  quantities,  if  r  denotes  the  probable  error  of  an  obser- 
vation of  the  weight  unity,  we  have  the  following  equations: — 

r  =  0.67449 
r 

=  i/? 

145.  The  formulae  which  result  from  the  theory  of  errors  according 
to  which  the  method  of  least  squares  is  derived,  enable  us  to  combine 
the  data  furnished  by  observation  so  as  to  overcome,  in  the  greatest 
degree  possible,  the  effect  of  those  accidental  errors  which  no  refine- 
ment of  theory  can  successfully  eliminate.  The  problem  of  the  cor- 
rection of  the  approximate  elements  of  the  orbit  of  a  heavenly  body 
by  means  of  a  series  of  observed  places,  requires  the  application  of 
nearly  all  the  distinct  results  which  have  been  derived.  The  first 
approximate  elements  of  the  orbit  of  the  body  will  be  determined 
om  three  or  four  observed  places  according  to  the  methods  which 

26 


102  THEORETICAL   ASTRONOMY. 

have  been  already  explained.  In  the  case  of  a  planet,  if  the  inclina- 
tion is  not  very  small,  the  method  of  three  geocentric  places  may  be 
employed,  but  it  will,  in  general,  afford  greater  accuracy  and  require 
but  little  additional  labor  to  base  the  first  determination  on  four 
observed  places,  according  to  the  process  already  illustrated.  In  the 
case  of  a  comet,  the  first  assumption  made  is  that  the  orbit  is  a 
parabola,  and  the  elements  derived  in  accordance  with  this  hypothesis 
may  be  successively  corrected,  until  it  is  apparent  whether  it  is  ne- 
cessary to  make  any  further  assumption  in  regard  to  the  value  of  the 
eccentricity.  In  all  cases,  the  approximate  elements  derived  from  a 
few  places  should  be  further  corrected  by  means  of  more  extended 
data  before  any  attempt  is  made  to  obtain  a  more  complete  determi- 
nation of  the  elements.  The  various  methods  by  which  this  pre- 
liminary correction  may  be  effected  have  been  already  sufficiently  de- 
veloped. 

The  fundamental  places  adopted  as  the  basis  of  the  correction  may 
be  single  observed  places  separated  by  considerable  intervals  of  time; 
but  it  will  be  preferable  to  use  places  which  may  be  regarded  as  the 
average  of  a  number  of  observations  made  on  the  same  day  or  during 
a  few  days  before  and  after  the  date  of  the  average  or  normal  place. 
The  ephemeris  computed  from  the  approximate  elements  known  may 
be  assumed  to  represent  the  actual  path  so  closely  that,  for  an  interval 
of  a  few  days,  the  difference  between  computation  and  observation 
may  be  regarded  as  being  constant,  or  at  least  as  varying  proportion- 
ally to  the  time.  Let  n,  n',  n",  &c.  be  the  differences  between  com- 
putation and  observation,  in  the  case  of  either  spherical  co-ordinate, 
for  the  dates  t,  tf,  tn  ',  &c.,  respectively;  then,  if  the  interval  between 
the  extreme  observations  to  be  combined  in  the  formation  of  the 
normal  place  is  not  too  great,  and  if  we  regard  the  observations  as 
equally  precise,  the  normal  difference  nQ  between  computation  and 
observation  will  be  found  by  taking  the  arithmetical  mean  of  the 
several  values  of  n,  and  this  being  applied  with  the  proper  sign  to 
the  computed  spherical  co-ordinate  for  the  date  £0,  which  is  the  mean 
of  t,  I',  t",  &c.j  will  give  the  corresponding  normal  place.  But  when 
different  weights  p,  p',  pff,  &c.  are  assigned  to  the  observations,  the 
value  of  n0  must  be  found  from 

_  np  -f  n'p'  +  n"p"  +  .  .  .  . 

> 


and  the  weight  of  this  value  will  be  equal  to  the  sum 
P+p' 


COMBINATION    OF   OBSERVATIONS.  403 

The  date  of  the  normal  place  will  be  determined  by 

_pt+P-e  +  prr  +  .... 
*      P+P'+P"  +  ....  ' 

]f  the  error  of  the  ephemeris  can  be  considered  as  nearly  constant, 
it  is  not  necessary  to  determine  £0  with  great  precision,  since  any  date 
not  differing  much  from  the  average  of  all  may  be  adopted  with  suf- 
ficient accuracy.  It  should  be  observed  further  that,  in  order  to 
obtain  the  greatest  accuracy  practicable,  the  spherical  co-ordinates  of 
the  body  for  the  date  t0  should  be  computed  directly  from  the  elements, 
so  that  the  resulting  normal  place  may  be  as  free  as  possible  from  the 
effect  of  neglected  differences  in  the  interpolation  of  the  ephemeris. 

When  the  differences  between  the  computed  and  the  observed 
places  to  be  combined  for  the  formation  of  a  normal  place  cannot  be 
considered  as  varying  proportionally  to  the  time,  we  may  derive  the 
error  of  the  ephemeris  from  an  equation  of  the  form  of  (53)6,  namely, 


the  coefficients  J.,  jB,  and  C  being  found  from  equations  of  condition 
formed  by  means  of  the  several  known  values  of  A#  in  the  case  of 
each  of  the  spherical  co-ordinates. 

146.  In  this  way  we  obtain  normal  places  at  convenient  intervals 
throughout  the  entire  period  during  which  the  body  was  observed. 
From  three  or  more  of  these  normal  places,  a  new  system  of  elements 
should  be  computed  by  means  of  some  one  of  the  methods  which 
have  already  been  given;  and  these  fundamental  places  being  judi- 
ciously selected,  the  resulting  elements  will  furnish  a  pretty  close 
approximation  to  the  truth,  so  that  the  residuals  which  are  found  by 
comparing  them  with  all  the  directly  observed  places  may  be  regarded 
as  indicating  very  nearly  the  actual  errors  of  those  places.  We  may 
then  proceed  to  investigate  the  character  of  the  observations  more 
fully.  But  since  the  observations  will  have  been  made  at  many  dif- 
ferent places,  by  different  observers,  with  instruments  of  different 
sizes,  and  under  a  variety  of  dissimilar  attendant  circumstances,  it 
may  be  easily  understood  that  the  investigation  will  involve  much 
that  is  vague  and  uncertain.  In  the  theory  of  errors  which  has  been 
developed  in  this  chapter,  it  has  been  assumed  that  all  constant 
errors  have  been  duly  eliminated,  and  that  the  only  errors  which 
remain  are  those  accidental  errors  which  must  ever  continue  in  a 
greater  or  less  degree  undetermined.  The  greater  the  number  and 


404  THEORETICAL    ASTRONOMY. 

perfection  of  the  observations  employed,  the  more  nearly  will  these 
errors  be  determined,  and  the  more  nearly  will  the  law  of  their  dis- 
tribution conform  to  that  which  has  been  assumed  as  the  basis  of 
the  method  of  least  squares. 

When  all  known  errors  have  been  eliminated,  there  may  yet  remain 
constant  errors,  and  also  other  errors  whose  law  of  distribution  is 
peculiar,  such  as  may  arise  from  the  idiosyncrasies  of  the  different 
observers,  from  the  systematic  errors  of  the  adopted  star-places'  in 
the  case  of  differential  observations,  and  from  a  variety  of  other 
sources;  and  since  the  observations  themselves  furnish  the  only  means 
of  arriving  at  a  knowledge  of  these  errors,  it  becomes  important  to 
discuss  them  in  such  a  manner  that  all  errors  which  may  be  regarded, 
in  a  sense  more  or  less  extended,  as  regular  may  be  eliminated. 
When  this  has  been  accomplished,  the  residuals  which  still  remain 
will  enable  us  to  form  an  estimate  of  the  degree  of  accuracy  which 
may  be  attributed  to  the  different  series  of  observations,  in  order  that 
they  may  not  only  be  combined  in  the  most  advantageous  manner, 
but  that  also  no  refinements  of  calculation  may  be  introduced  which 
are  not  warranted  by  the  quality  of  the  material  to  be  employed. 

The  necessity  of  a  preliminary  calculation  in  which  a  high  degree 
of  accuracy  is  already  obtained,  is  indicated  by  the  fact  that,  however 
conscientious  the  observer  may  be,  his  judgment  is  unconsciously 
warped  by  an  inherent  desire  to  produce  results  harmonizing  well 
among  themselves,  so  that  a  limited  series  of  places  may  agree  to 
such  an  extent  that  the  probable  error  of  an  observation  as  derived 
from  the  relative  discordances  would  assign  a  weight  vastly  in  excess 
of  its  true  value.  The  combination,  however,  of  a  large  number  of 
independent  data,  by  exhibiting  at  least  an  approximation  to  the 
absolute  errors  of  the  observations,  will  indicate  nearly  what  the 
measure  of  precision  should  be.  As  soon,  therefore,  as  provisional 
elements  which  nearly  represent  the  entire  series  of  observations  have 
been  found,  an  attempt  should  be  made  to  eliminate  all  errors  which 
may  be  accurately  or  approximately  determined.  The  places  of  the 
comparison-stars  used  in  the  observations  should  be  determined  with 
care  from  the  data  available,  and  should  be  reduced,  by  means  of  the 
proper  systematic  corrections,  to  some  standard  system.  The  reduc- 
tion of  the  mean  places  of  the  stars  to  apparent  places  should  also  be 
made  by  means  of  uniform  constants  of  reduction.  The  observations 
will  thus  be  uniformly  reduced.  Then  the  perturbations  arising  from 
the  action  of  the  planets  should  be  computed  by  means  of  formula 
which  will  be  investigated  in  the  next  chapter,  and  the  observed 


COMBINATION   OF   OBSERVATIONS.  405 

places  should  be  freed  from  these  perturbations  so  as  to  give  the 
places  for  a  system  of  osculating  elements  for  a  given  date. 

147.  The  next  step  in  the  process  will  be  to  compare  the  pro- 
visional elements  with  the  entire  series  of  observed  places  thus  cor- 
rected; and  in  the  calculation  of  the  ephemeris  it  will  be  advan- 
tageous to  correct  the  places  of  the  sun  given  by  the  tables  whenever 
observations  are  available  for  that  purpose.  Then,  selecting  one  or 
more  epochs  as  the  origin,  if  we  compute  the  coefficients  A,  B,  C  in 
the  equation 

A0  =  A  +  Br  +  Cr\  (125) 

in  the  case  of  each  of  the  spherical  co-ordinates,  by  means  of  equa- 
tions of  condition  formed  from  all  the  observations,  the  standard 
ephemeris  may  be  corrected  so  that  it  may  be  regarded  as  representing 
the  actual  path  of  the  body  during  the  period  included  by  the  obser- 
vations. When  the  number  of  observations  is  sonsiderable,  it  will  be 
more  convenient  to  divide  the  observations  into  groups,  and  use  the 
differences  between  computation  and  observation  for  provisional 
normal  places  in  the  formation  of  the  equations  of  condition  for  the 
determination  of  A,  B,  and  C.  It  thus  appears  that  the  corrected 
ephemeris  which  is  so  essential  to  a  determination  of  the  constant 
errors  peculiar  to  each  series  of  observations,  is  obtained  without  first 
having  determined  the  most  probable  system  of  elements.  The  cor- 
rections computed  by  means  of  the  equation  (125)  being  applied  to 
the  several  residuals  of  each  series,  we  obtain  what  may  be  regarded 
as  the  actual  errors  of  these  observations.  The  arithmetical  or  pro- 
bable mean  of  the  corrected  residuals  for  the  series  of  observations 
made  by  each  observer  may  be  regarded  as  the  average  error  of  obser- 
vation for  that  series.  The  mean  of  the  average  errors  of  the  several 
series  may  be  regarded  as  the  actual  constant  error  pertaining  to  all 
the  observations,  and  the  comparison  of  this  final  mean  with  the 
means  found  for  the  different  series,  respectively,  furnishes  the  pro- 
bable value  of  the  constant  errors  due  to  the  peculiarities  of  the 
observers;  and  the  constant  correction  thus  found  for  each  observer 
should  be  applied  to  the  corresponding  residuals  already  obtained. 

In  this  investigation,  if  the  number  of  comparisons  or  the  number 
of  wires  taken  is  known,  relative  weights  proportional  to  the  number 
of  comparisons  may  be  adopted  for  the  combination  of  the  residuals 
for  each  series.  In  this  manner,  observations  which,  on  account  of 
the  peculiarities  of  the  observers,  are  in  a  certain  sense  heterogeneous, 
may  be  rendered  homogeneous,  being  reduced  to  a  standard  which 


406  THEORETICAL   ASTRONOMY. 

approaches  the  absolute  in  proportion  as  the  number  and  perfection 
of  the  distinct  series  combined  are  increased.  Whatever  constant 
error  remains  will  be  very  small,  and,  besides,  will  affect  all  places 
alike. 

The  residuals  which  now  remain  must  be  regarded  as  consisting 
of  the  actual  errors  of  observation  and  of  the  error  of  the  adopted 
place  of  the  comparison-star.  Hence  they  will  not  give  the  probable 
error  of  observation,  and  will  not  serve  directly  for  assigning  the 
measures  of  precision  of  the  series  of  observations  by  each  observer. 
Let  us,  therefore,  denote  by  e,  the  mean  error  of  the  place  of  the 
comparison-star,  by  e,  the  mean  error  of  a  single  comparison;  then 

will  —4=-  be  the  mean  error  of  m  comparisons,  and  the  mean  error  of 

V  m 
the  resulting  place  of  the  body  will,  according  to  equation  (35),  be 

given  by 

•.•  =  £  +  '.'.  (126) 

The  value  of  e0,  in  the  case  of  each  series,  will  be  found  by  means  of 
the  residuals  finally  corrected  for  the  constant  errors,  and  the  value 
of  es  is  supposed  to  be  determined  in  the  formation  of  the  catalogue 
of  star-places  adopted.  Hence  the  actual  mean  error  of  an  observa- 
tion consisting  of  a  single  comparison  will  be 


e,  =  l/m(>02-e,2).  (127) 

The  value  of  e,  for  each  observer  having  been  found  in  accordance 
with  this  equation,  the  mean  error  of  an  observation  consisting  of  m 
comparisons  will  be 

Sf 

Vm 

The  mean  error  of  an  observation  whose  weight  is  unity  being  de- 
noted by  e,  the  weight  of  an  observation  based  on  m  comparisons  will 
be 

,  =  £  (128) 

The  value  of  e  may  be  arbitrarily  assigned,  and  we  may  adopt  for  it 
±  10"  or  any  other  number  of  seconds  for  which  the  resulting  values 
of  p  will  be  convenient  numbers. 

When  all  the  observations  are  differential  observations,  and  the  stars 
of  comparison  are  included  in  the  fundamental  list,  if  we  do  not  take 
into  account  the  number  of  comparisons  on  which  each  observed 


COMBINATION   OF   OBSERVATIONS.  407 

place  depends,  it  will  not  be  necessary  to  consider  et)  and  we  may 
then  derive  e,  directly  from  the  residuals  corrected  for  constant  errors. 
Further,  in  the  case  of  meridian  observations,  the  error  which  corre- 
sponds to  es  will  be  extremely  small,  and  hence  it  is  only  when  these 
are  combined  with  equatorial  observations,  or  when  equatorial  obser- 
vations based  on  different  numbers  of  comparisons  are  combined,  that 
the  separation  of  the  errors  into  the  two  component  parts  becomes 
necessary  for  a  proper  determination  of  the  relative  weights. 

According  to  the  complete  method  here  indicated,  after  having 
eliminated  as  far  as  possible  all  constant  errors,  including  the  correc- 
tions assigned  by  equation  (125)  to  be  applied  to  the  provisional 
ephemeris,  we  find  the  value  of  e,  given  by  the  equation 

ne,*  =  [row]  —  [m]  e,2,  (129) 

in  which  n  denotes  the  number  of  observations;  m,  m',  m",  &c.  the 
number  of  comparisons  for  the  respective  observations;  and  v,  v',  vff, 
&c.  the  corresponding  residuals.  Then,  by  means  of  equation  (128), 
assuming  a  convenient  number  for  e,  we  compute  the  weight  of  each 
observation.  Thus,  for  example,  let  the  residuals  and  corresponding 
values  of  m  be  as  follows  :  — 

Ad  m  Ad  m 

+  2".0  5,  —  1".0  7, 

—  1  .8  5,  +  1  .5  5, 

—  0  .4  10,  +4  .1  8, 

—  5  .5  5,                            0  .0  5. 

Let  the  mean  error  of  the  place  of  a  comparison-star  be 


then  we  have  n  ~—  8,  and,  according  to  (129), 

8s,2  =341.78  —  200.0, 
which  gives 

*,=  :±4".2. 

Let  us  now  adopt  as  the  unit  of  weight  that  for  which  the  mean  erroi  is 


then  we  obtain  by  means  of  equation  (128),  for  the  weights  of  the 
observations, 

2.5,        2.5,        5.1,        2.5,        3.6,        2.5,        4.1,        2.5, 

respectively.  • 


408  THEORETICAL   ASTRONOMY. 

In  this  manner  the  weights  of  the  observations  in  the  series  made 
by  each  observer  must  be  determined,  using  throughout  the  same 
value  of  e.  Then  the  differences  between  the  places  computed  from 
the  provisional  elements  to  be  corrected  and  the  observed  places  cor- 
rected for  the  constant  error  of  the  observer,  must  be  combined  ac- 
cording to  the  equations  (123)  and  (125),  the  adopted  values  of  p,  p', 
p"j  &c.  being  those  found  from  (128).  Thus  will  be  obtained  the 
final  residuals  for  the  formation  of  the  equations  of  condition  from 
which  to  derive  the  most  probable  value  of  the  corrections  to  be 
applied  to  the  elements.  The  relative  weights  of  these  normals  will 
be  indicated  by  the  sums  formed  by  adding  together  the  weights  of 
the  observations  combined  in  the  formation  of  each  normal,  and  the 
unit  of  weight  will  depend  on  the  adopted  value  of  e.  If  it  be  de- 
sired to  adopt  a  different  unit  of  weight  in  the  case  of  the  solution 
of  the  equations  of  condition,  such,  for  example,  that  the  weight  of 
an  equation  of  average  precision  shall  be  unity,  we  may  simply  divide 
the  weights  of  the  normals  by  any  number  p0  which  will  satisfy  the 
condition  imposed.  The  mean  error  of  an  observation  whose  weight 
is  unity  will  then  be  given  by 


the  value  of  e  being  that  used  in  the  determination  of  the  weights  p, 
p',  &o. 

148.  The  observations  of  comets  are  liable  to  be  affected  by  other 
errors  in  addition  to  those  which  are  common  to  these  and  to  planet- 
ary observations.  Different  observers  will  fix  upon  different  points 
as  the  proper  point  to  be  observed,  and  all  of  these  may  differ  from 
the  actual  position  of  the  centre  of  gravity  of  the  comet;  and  fur- 
ther, on  account  of  changes  in  the  physical  appearance  of  the  comet, 
the  same  observer  may  on  different  nights  select  different  points. 
These  circumstances  concur  to  vitiate  the  normal  places,  inasmuch  as 
the  resulting  errors,  although  in  a  certain  sense  fortuitous,  are  yet 
such  that  the  law  of  their  distribution  is  evidently  different  from 
that  which  is  adopted  as  the  basis  of  the  method  of  least  squares. 
The  impossibility  of  assigning  the  actual  limits  and  the  law  of  dis- 
tribution of  many  errors  of  this  class,  renders  it  necessary  to  adopt 
empirical  methods,  the  success  of  which  will  depend  on  the  discrimi- 
nation of  the  computer. 

If  e0  denotes  the  mean  jerror  of  an  observation  based  on  m  com- 


COMBINATION   OF   OBSERVATIONS.  409 

parisons,  and  ec  the  mean  error  to  be  feared  on  account  of  the  pecu- 
liarities of  the  physical  appearance  of  the  comet, 


will  express  the  mean  error  of  the  residuals;  and  if  n  of  these 
residuals  are  combined  in  the  formation  of  a  normal  place,  the  mean 
error  of  the  normal  will  be  given  by 

en'  =  M -{-./.  (130) 

The  value  of  ec2  may  be  determined  approximately  from  the  data 
furnished  by  the  observations.  Thus,  if  the  mean  error  of  a  single 
comparison,  for  the  different  observers,  has  been  determined  by  means 
of  the  differences  between  single  comparisons  and  the  arithmetical 
mean  of  a  considerable  number  of  comparisons,  and  if  the  mean  error 
of  the  place  of  a  comparison-star  has  also  been  determined,  the 
equation  (126)  will  give  the  corresponding  value  of  £02;  then  the 
actual  differences  between  computation  and  observation  obtained  by 
eliminating  the  error  of  the  ephemeris  and  such  constant  errors  as 
may  be  determined,  will  furnish  an  approximate  value  of  ec  by  means 
of  the  formula 


in  which  n  denotes  the  number  of  observations  combined. 

Sometimes,  also,  in  the  case  of  comets,  in  order  to  detect  the  opera~ 
tion  of  any  abnormal  force  or  circumstance  producing  different  effects 
in  different  parts  of  the  orbit,  it  may  be  expedient  to  divide  the 
observations  into  two  distinct  groups,  the  first  including  the  observa- 
tions made  before  the  time  of  perihelion  passage,  and  the  other 
including  those  subsequent  to  that  epoch. 

149.  The  circumstances  of  the  problem  will  often  suggest  appro- 
priate modifications  of  the  complete  process  of  determining  the  rela- 
tive weights  of  the  observations  to  be  combined,  or  indeed  a  relaxa- 
tion from  the  requirements  of  the  more  rigorous  method.  Thus,  if 
on  account  of  the  number  or  quality  of  the  data  it  is  not  considered 
necessary  to  compute  the  relative  weights  with  the  greatest  precision 
attainable,  it  will  suffice,  when  the  discussion  of  the  observations  has 
been  carried  to  an  extent  sufficient  to  make  an  approximate  estimate 
of  the  relative  weights,  to  assume,  without  considering  the  number 
of  comparisons,  a  weight  1  for  the  observations  at  one  observatory,  a 


410  THEORETICAL   ASTRONOMY. 

weight  |  for  another  class  of  observations,  J  for  a  third  class,  and  so 
on.  It  should  be  observed,  also,  that  when  there  are  but  few  obser- 
vations to  be  combined,  the  application  of  the  formulae  for  the  mean 
or  probable  errors  may  be  in  a  degree  fallacious,  the  resulting  values 
of  these  errors  being  little  more  than  rude  approximations ;  still  the 
mean  or  probable  errors  as  thus  determined  furnish  the  most  reliable 
means  of  estimating  the  relative  weights  of  the  observations  made 
by  different  observers,  since  otherwise  the  scale  of  weights  would 
depend  on  the  arbitrary  discretion  of  the  computer.  Further,  in  a 
complete  investigation,  even  when  the  very  greatest  care  has  been 
taken  in  the  theoretical  discussion,  on  account  of  independent  known 
circumstances  connected  with  some  particular  observation,  it  may  be 
expedient  to  change  arbitrarily  the  weight  assigned  by  theory  to 
certain  of  the  normal  places.  It  may  also  be  advisable  to  reject 
entirely  those  observations  whose  weight  is  less  than  a  certain  limit 
which  may  be  regarded  as  the  standard  of  excellence  below  which 
the  observations  should  be  rejected;  and  it  will  be  proper  to  reject 
observations  which  do  not  afford  the  data  requisite  for  a  homogeneous 
combination  with  the  others  according  to  the  principles  already 
explained.  But  in  all  cases  the  rejection  of  apparently  doubtful 
observations  should  not  be  carried  to  any  considerable  extent  unless 
a  very  large  number  of  good  observations  are  available.  The  mere 
apparent  discrepancy  between  any  residual  and  the  others  of  a  series, 
is  not  in  itself  sufficient  to  warrant  its  rejection  unless  facts  are 
known  which  would  independently  assign  to  it  a  low  degree  of  pre- 
cision. 

A  doubtful  observation  will  have  the  greatest  influence  in  vitiating 
the  resulting  normal  place  when  but  a  small  number  of  observed 
places  are  combined ;  and  hence,  since  we  cannot  assume  that  the  law 
of  the  distribution  of  errors,  according  to  which  the  method  of  least 
squares  is  derived,  will  be  complied  with  in  the  case  of  only  a  few 
observations,  it  will  not  in  general  be  safe  to  reject  an  observation  pro- 
vided that  it  surpasses  a  limit  which  is  fixed  by  the  adopted  theory 
of  errors.  If  the  number  of  observations  is  so  large  that  the  dis- 
tribution of  the  errors  may  be  assumed  to  conform  to  the  theory 
adopted,  it  will  be  possible  to  assign  a  limit  such  that  a  residual 
which  surpasses  it  may  be  rejected.  Thus,  in  a  series  of  m  observa- 
tions, according  to  the  expression  (19),  the  number  of  errors  greater 
than  nr  will  be 


COMBINATION   OF   OBSERVATIONS. 


4J1 


and  when  n  has  a  value  such  that  the  value  of  this  expression  is  less 
than  0.5,  the  error  nr  will  have  a  greater  probability  against  it  than 
for  it,  and  hence  it  may  be  rejected.  The  expression  for  finding  the 
limiting  value  of  n  therefore  becomes 


2m 


(131) 


By  means  of  this  equation  we  derive  for  given  values  of  m  the  cor- 
responding values  of  nhr  =  0.47694n,  and  hence  the  values  of  n. 
For  convenient  application,  it  will  be  preferable  to  use  e  instead  of  r, 
and  if  we  put  nf  =  0.67449w,  the  limiting  error  will  be  n'e,  and  the 
values  of  n'  corresponding  to  given  values  of  m  will  be  as  exhibited 
in  the  following  table. 

TABLE. 


m 

«' 

m 

n' 

m 

n' 

m 

„'   | 

i 

6 

1.732 

20 

2.241 

55 

2.608 

90 

2.773 

8 

1.863 

25 

2.326 

60 

2.638 

95 

2.791 

10 

1.960 

30 

2.394 

65 

2.665 

100 

2.807 

12 

2.037 

35 

2.450 

70 

2.690 

200 

3.020 

14 

2.100 

40 

2.498 

75 

2.713 

300 

3.143 

16 

2.154 

45 

2.539 

80 

2.734 

400 

3.224 

18 

2.200 

50 

2.576 

85 

2.754 

500 

3.289 

According  to  this  method,  we  first  find  the  mean  error  of  an  obser- 
vation by  means  of  all  the  residuals.  Then,  with  the  value  of  m  as 
the  argument,  we  take  from  the  table  the  corresponding  value  of  n', 
and  if  one  of  the  residuals  exceeds  the  value  n'e  it  must  be  rejected, 
Again,  finding  a  new  value  of  e  from  the  remaining  m  —  1  residuals, 
and  repeating  the  operation,  it  will  be  seen  whether  another  observa- 
tion should  be  rejected ;  and  the  process  may  be  continued  until  a 
limit  is  reached  which  does  not  require  the  further  rejection  of  ob- 
servations. Thus,  for  example,  in  the  case  of  50  observations  in 
which  the  residuals  — 11". 5  and  +7". 8  occur,  let  the  sum  of  the 
squares  of  the  residuals  be 

[w]  =  320.4. 

Then,  according  to  equation  (30),  we  shall  have 

*  =  ±  2".56. 


412  THEORETICAL   ASTRONOMY. 

Corresponding  to  the  value  m  =  50,  the  table  gives  nf  --  2.576,  and 
the  limiting  value  of  the  error  becomes 


and  hence  the  residuals  —  ll/7.5  and  +  7".8  are  rejected.     Recom- 
puting the  mean  error  of  an  observation,  we  have 


320-4 -193.0S    =±1".65. 

In  the  formation  of  a  normal  place,  when  the  mean  error  of  an 
observation  has  been  inferred  from  only  a  small  number  of  observa- 
tions, according  to  what  has  been  stated,  it  will  not  be  safe  to  rely 
upon  the  equation  (131)  for  the  necessity  of  the  rejection  of  a  doubt- 
ful observation.  But  if  any  abnormal  influence  is  suspected,  or  if 
any  antecedent  discussion  of  observations  by  the  same  observer,  made 
under  similar  circumstances,  seems  to  indicate  that  an  error  of  a  given 
magnitude  is  highly  improbable,  the  application  of  this  formula  will 
serve  to  confirm  or  remove  the  doubt  already  created.  Much  will 
therefore  depend  on  the  discrimination  of  the  computer,  and  on  his 
knowledge  of  the  various  sources  of  error  which  may  conspire  con- 
tinuously or  discontinuously  in  the  production  of  large  apparent 
errors.  It  is  the  business  of  the  observer  to  indicate  the  circum- 
stances peculiar  to  the  phenomenon  observed,  the  instruments  em- 
ployed, and  the  methods  of  observation;  and  the  discussion  of  the 
data  thus  furnished  by  different  observers,  as  far  as  possible  in  ac- 
cordance with  the  strict  requirements  of  the  adopted  theory  of  errors, 
will  furnish  results  which  must  be  regarded  as  the  best  which  can  be 
derived  from  the  evidence  contributed  by  all  the  observations. 

150.  When  the  final  normal  places  have  been  derived,  the  differ- 
ences between  these  and  the  corresponding  places  computed  from  the 
provisional  elements  to  be  corrected,  taken  in  the  sense  computation 
minus  observation,  give  the  values  of  n,  nf,  n"}  &c.  which  are  the 
absolute  terms  of  the  equations  of  condition.  By  means  of  these 
elements  we  compute  also  the  values  of  the  differential  coefficients  of 
each  of  the  spherical  co-ordinates  with  respect  to  each  of  the  elements 
to  be  corrected.  These  differential  coefficients  give  the  values  of  the 
coefficients  a,  6,  c,  a',  &',  &c.  in  the  equations  of  condition.  The 
mode  of  calculating  these  coefficients,  for  different  systems  of  co-or- 
dinates, and  the  mode  of  forming  the  equations  of  condition,  have 
been  fully  developed  in  the  second  chapter.  It  is  of  great  import- 


CORRECTION   OF   THE   ELEMENTS.  413 

ance  that  the  numerical  values  of  these  coefficients  should  be  care- 
fully checked  by  direct  calculation,  assigning  variations  to  the  ele- 
ments, or  by  means  of  differences  when  this  test  can  be  successfully 
applied.  In  assigning  increments  to  the  elements  in  order  to  check 
the  formation  of  the  equations,  they  should  not  be  so  large  that  the 
neglected  terms  of  the  second  order  become  sensible,  nor  so  small  that 
they  do  not  afford  the  required  certainty  by  means  of  the  agreement 
of  the  corresponding  variations  of  the  spherical  co-ordinates  as 
obtained  by  substitution  and  by  direct  calculation. 

As  soon  as  the  equations  of  condition  have  been  thus  formed,  we 
multiply  each  of  them  by  the  square  root  of  its  weight  as  given  by 
the  adopted  relative  weights  of  the  normal  places;  and  these  equa- 
tions will  thus  be  reduced  to  the  same  weight.  In  general,  tho 
numerical  values  of  the  coefficients  will  be  such  that  it  will  be  con- 
venient, although  not  essential,  to  adopt  as  the  unit  of  weight  that 
which  is  the  average  of  the  weights  of  the  normals,  so  that  the 
numbers  by  which  most  of  the  equations  will  be  multiplied  will  not 
differ  much  from  unity.  The  reduction  of  the  equations  to  a  uniform 
measure  of  precision  having  been  effected,  it  remains  to  combine  them 
according  to  the  method  of  least  squares  in  order  to  derive  the  most 
probable  values  of  the  unknown  quantities,  together  with  the  relative 
weights  of  these  values.  It  should  be  observed,  however,  that  the 
numerical  calculation  in  the  combination  and  solution  of  these  equa- 
tions, and  especially  the  required  agreement  of  some  of  the  checks  of 
the  calculation,  will  be  facilitated  by  having  the  numerical  values  of 
the  several  coefficients  not  very  unequal.  If,  therefore,  the  coefficient 
a  of  any  unknown  quantity  x  is  in  each  of  the  equations  numerically 
much  greater  or  much  less  than  in  the  case  of  the  other  unknown 
quantities,  we  may  adopt  as  the  corresponding  unknown  quantity  to 
be  determined,  not  x  but  vz,  i/  being  any  entire  or  fractional  number 

such  that  the  new  coefficients  -,  T,  &c.  shall  be  made  to  agree  in 

magnitude  with  the  other  coefficients.  The  unknown  quantity  whose 
value  will  then  be  derived  by  the  solution  of  the  equations  will  be 
vx,  and  the  corresponding  weight  will  be  that  of  vx.  To  find  the 
weight  of  x  from  that  of  vx,  we  have  the  equation 

?.  =  >*-.  (132) 

In  the  same  manner,  the  coefficient  of  any  other  unknown  quantity 
may  be  changed,  and  the  coefficients  of  all  the  unknown  quantities 
may  thus  be  made  to  agree  in  magnitude  within  moderate  limits,  th* 


414  THEORETICAL   ASTRONOMY. 

advantage  of  which,  in  the  numerical  solution  of  the  equations,  will 
be  apparent  by  a  consideration  of  the  mode  of  proving  the  calcula- 
tion of  the  coefficients  in  the  normal  equations.  It  will  be  expedient, 
also,  to  take  for  v  some  integral  power  of  10,  or,  when  a  fractional 
value  is  required,  the  corresponding  decimal.  It  may  be  remarked, 
further,  that  the  introduction  of  v  is  generally  required  only  when 
the  coefficient  of  one  of  the  unknown  quantities  is  very  large,  as 
frequently  happens  in  the  case  of  the  variation  of  the  mean  daily 
motion  //. 

When  the  coefficients  of  some  of  the  unknown  quantities  are 
extremely  small  in  all  the  equations  of  condition  to  be  combined,  an 
approximate  solution,  and  often  one  which  is  sufficiently  accurate  for 
the  purposes  required,  may  be  obtained  by  first  neglecting  these 
quantities  entirely,  and  afterwards  determining  them  separately.  In 
general,  however,  this  can  only  be  done  when  it  is  certainly  known 
that  the  influence  of  the  neglected  terms  is  not  of  sensible  magnitude, 
or  when  at  least  approximate  values  of  these  terms  are  already  given. 
When  we  adopt  the  approximate  plane  of  the  orbit  as  the  funda- 
mental plane,  the  equations  for  the  longitude  involve  only  four  ele- 
ments, and  the  coefficients  of  the  variations  of  these  elements  in  the 
equations  for  the  latitudes  are  always  very  small.  Hence,  for  an 
approximate  solution,  we  may  first  solve  the  equations  involving  four 
unknown  quantities  as  furnished  by  the  longitudes,  and  then,  substi- 
tuting the  resulting  values  in  the  equations  for  the  latitudes,  they 
will  contain  but  two  unknown  quantities,  namely,  those  which  give 
the  corrections  to  be  applied  to  &  and  i. 

151.  When  the  number  of  equations  of  condition  is  large,  the 
computation  of  the  numerical  values  of  the  coefficients  in  the  normal 
equations  will  entail  considerable  labor;  and  hence  it  is  desirable  to 
arrange  the  calculation  in  a  convenient  form,  applying  also  the  checks 
which  have  been  indicated.  The  most  convenient  arrangement  will 
be  to  write  the  logarithms  of  the  absolute  terms  n,  nf,  n",  &c.  in  a 
horizontal  line,  directly  under  these  the  logarithms  of  the  coefficients 
a,  a',  a",  &c.,  then  the  logarithms  of  6,  &',  6",  &c.,  and  so  on.  Then 
writing,  in  a  corresponding  form,  the  values  of  logn,  lognr,  &c.  on  a 
slip  of  paper,  by  bringing  this  successively  over  each  line,  the  sums 
[nri],  [an],  [6n],  &c.  will  be  readily  formed.  Again,  writing  on 
another  slip  of  paper  the  logarithms  of  a,  a',  a",  &c.,  and  placing 
this  slip  successively  over  the  lines  containing  the  coefficients,  we 
derive  the  values  [aa],  [a&],  [ac],  &c.  The  multiplication  by  6,  c,  dt 


CORRECTION   OF   THE   ELEMENTS.  415 

&c.  successively  is  effected  in  a  similar  manner;  and  thus  will  be 
derived  [66],  [6c],  [bcl],  &c.,  and  finally  \_ff]  in  the  case  of  six  un- 
known quantities.  In  forming  these  sums,  in  the  cases  of  sums  of 
positive  and  negative  quantities,  it  is  convenient  as  well  as  conducive 
to  accuracy  to  write  the  positive  values  in  one  vertical  column  and 
the  negative  values  in  a  separate  column,  and  take  the  difference  of 
the  sums  of  the  numbers  in  the  respective  columns.  The  proof  of 
the  calculation  of  the  coefficients  of  the  normal  equations  is  effected 
by  introducing  s,  s',  s",  &c.,  the  algebraic  sums  of  all  the  coefficients 
in  the  respective  equations  of  condition,  and  treating  these  as  the 
coefficients  of  an  additional  unknown  quantity,  thus  forming  directly 
the  sums  [sn],  [as],  [6s],  [cs],  &c.  Then,  according  to  the  equations 
(76)  and  (77),  the  values  thus  found  should  agree  with  those  obtained 
by  taking  the  corresponding  sums  of  the  coefficients  in  the  normal 
equations. 

The  normal  equations  being  thus  derived,  the  next  step  in  the 
process  is  the  determination  of  the  values  of  the  auxiliary  quantities 
necessary  for  the  formation  of  the  equations  (74).  An  examination 
of  the  equations  (54),  (55),  &c.,  by  means  of  which  these  auxiliaries 
are  determined,  will  indicate  at  once  a  convenient  and  systematic 
arrangement  of  the  numerical  calculation.  Thus,  we  first  write  in  a 
horizontal  line  the  values  of  [aa],  [a6],  [ac],  .  .  .  [as],  [an],  and  di- 
rectly under  them  the  corresponding  logarithms.  Next,  we  write 
under  these,  commencing  with  [a6],  the  values  of  [66],  [6c],  [bd], 

. .  [6s],  [6n] ;  then,  adding  the  logarithm  of  the  factor  •= — -  to  the 

\_aa\ 

logarithms  of  [a6],  [ac],  &c.  successively,  we  write   the  value  of 

p—=r  [a6]  under  [66],  that  of  p — =r  [ac]  under  [6c],  and  so  on.  Sub- 
[oa]  L  J  [aa]  L  J 

tracting  the  numbers  in  this  line  from  those  in  the  line  above,  the 
differences  give  the  values  of  [66.1],  [6c.l],  .  .  .  [6s.l],  [6n.l],  to  be 
written  in  the  next  line,  and  the  logarithms  of  these  we  write  directly 
under  them.  Then  we  write  in  a  horizontal  line  the  values  of  [cc], 
[cd], .  .  [cs],  [en],  placing  [cc]  under  [6c.l],  and,  having  added  the 

logarithm  of  ^ — -  to  the  logarithms  of  [ac],  [ad],  &c.  in  succession, 

we  derive,  according  to  the  equations  (55)  and  (58),  the  values  of 
[cc.l],  [coM], . .  [cs.l],  [cn.l],  which  are  to  be  placed  under  the  cor- 
responding quantities  [cc],  [cd],  &c.  Next,  we  subtract  from  these, 
respectively,  the  products  t 

[6c-1]r6on      [6e-1]rwn     [ic-1]rfcn      [6^r6»n 

f6UJ  [6<a]>        [663]  Lftrf-1J> '  '  [663]  L*S'1J'        f 66.11 L        ' 


416  THEORETICAL   ASTRONOMY. 

and  thus  derive  the  values  of  [cc.2],  [cd2],  .  .  [es.2],  [cn.2],  which 
are  to  be  written  in  the  next  horizontal  line  and  under  them  their 
logarithms.  Then  we  introduce,  in  a  similar  manner,  the  coefficients 
[dd],  [de]y  .  .  [dn],  writing  \_dd~]  under  [ccZ.2]  ;  and  from  each  of  these 
in  succession  we  subtract  the  products 


thus  finding  the  values  of  [<M.l],  [c?e.l],  .  .  [dn  .  1].     From  these  we 
subtract  the  products 


] 

respectively,  which  operation  gives  the  values  of  [dd2],  [efe.2],  .  .  . 
[c?n.2].     From  these  results  we  subtract  the  products 


W21  .  . 

[cc.2]  L<  [cc.2J  L<    ^J'  '  '  [cc.2]  L 

and  derive  [dd.3],  [de.3],  .  .  [cfri.3]  under  which  we  write  the  cor- 
responding logarithms.     Then  we  introduce  [ee]9  [ef~],  [es]9  and  [eri], 

writing  [ee]  under  [rfe.3].     First,  subtracting  p  —  =-  [ae],  p  —  -  [a/],  .  . 

[ae] 

f  —  r[a7i]>  we  ge^  [ee'^]j  [^/-l])  [6S'1]>  an(i  [0W-.1];  then  subtracting 


from  these  the  products 


j  '  '  [66.1] 

we  obtain  the  values  of  [ee.2],  [e/.2],  [es.2],  and    [en.2].     Again, 
subtracting 


we  have  the  values  of  [ee.3],  [^f.3],  [es.3],  [cn.3];  and  finally,  sub- 
tracting from  these  the  products 


j  '  '  E3Z3]      '> 

we  derive  the  results  for  [66.4],  [e/.4],  [es.4],  and  [en.4];  under  which 
the  corresponding  logarithms  are  to  be  written. 

If  there  are  six  unknown  quantities  to  be  determined,  we  must 
further  write  in  a  horizontal  line  the  values  of  [jj/*],  [/s],  and  [/n], 


CORRECTION   OF   THE   ELEMENTS.  41* 

placing  [jgf]  under  [e/.4],  and  by  means  of  five  successive  subtrac- 
tions entirely  analogous  to  what  precedes,  and  as  indicated  by  the 
remaining  equations  for  the  auxiliaries,  we  obtain  the  values  of  [j^.5], 
[/a.5],  and  [/n.5]. 

The  values  of  [fo.l],  [cs.l],  [cs.2],  &c.  serve  to  check  the  calcula- 
tion of  the  successive  auxiliary  coefficients.     Thus  we  must  have 

[56.1]  +  [6c.l]  -f-  [fcd.1]  +  [6e.l]  +  [6/.1]  =  [6s.l] 
[6c.l]  -f  [cc.l]  +  [cdl]  -f  [ce.l]  -f  [c/.l]  =  [«.l],  &c., 
[cc.2]  +  [ed.2]  -f  [ce.2]  -f  [c/.2]  =  [cs.2], 
[cd.2]  +  [eta.2]  +  [efe.2]  +  [d/.2]  =  [cfo.2],  Ac. 


Hence  it  appears  that  when  the  numerical  calculation  is  arranged  as 
above  suggested,  the  auxiliary  containing  s  must,  in  each  line,  be 
equal  to  the  sum  of  all  the  terms  to  the  left  of  it  in  the  same  line 
and  of  those  terms  containing  the  same  distinguishing  numeral  found 
in  a  vertical  column  over  the  last  quantity  at  the  left  of  this  line. 

There  will  yet  remain  only  the  auxiliaries  which  are  derived  from 
\_sn~]  and  [nri]  to  be  determined.  These  additional  auxiliaries  will 
be  found  by  means  of  the  formulae 


.  ,  -  . 

.3]  =  [m.2]  -  [es.2],         [m.4]  =  [m.3]  -  [efe.3],  (133) 


0».5]  =  [w.4]  -  [«.4],        [w.8]  =  [w.5]  - 

and  the  equations  (81)  and  (83).     The  arrangement  of  the  numerical 
process  should  be  similar  to  that  already  explained. 

The  values  of  [sw.l],  [sw.2],  &c.  check  the  accuracy  of  the  results 
for  [6n.l],  [cn.l],  [cn.2],  [e?n.3],  &c.  by  means  of  the  equations 

[bn.l]  +  [cn.l]  -f  \dn.V\  +  [67i.l]  +  [>.l]  =  [«i.l], 
[c7i.2]  +  [dn.Z]  -f-  [e^.2]  -f  [/7i.2]  ==  [sw.2], 

[dw.3]  -}-  [e».3]  +  LM]  =  [m.3],       (134; 


It  appears  further,  that,  in  the  case  of  six  unknown  quantities,  sinct 
[/s.5]  =  [jgr.5],  we  have  [«n.6]  =  0. 

Having  thus  determined  the  numerical  values  of  the  auxiliaries 
required,  we  are  prepared  to  form  at  once  the  equations  (74),  by  means 
of  which  the  values  of  the  unknown  quantities  will  be  determined 

27 


418  THEORETICAL   ASTRONOMY. 

by  successive  substitution,  first  finding  t  from  the  last  of  these  equa- 
tions, then  substituting  this  result  in  the  equation  next  to  the  last 
and  thus  deriving  the  value  of  w,  and  so  on  until  all  the  unknown 
quantities  have  been  determined.  It  will  be  observed  that  the  loga- 
rithms of  the  coefficients  of  the  unknown  quantities  in  these  equa- 
tions will  have  been  already  found  in  the  computation  of  the  aux- 
iliaries. 

If  we  add  together  the  several  equations  of  (74),  first  clearing  them 
of  fractions,  we  get 

0  =  [ad]  x  +  ([06]  +  [56.1])  y  +  ([oc]  +  [6c.l]  -f  [ee.2])  z 
+  (M  +  [W-l]  + 

+  (M  +  [&e-i]  + 

-f  ([«/]  +  P/1]  + 

+   [aw]  +  [6n.l]  +  [m.2]  -+  [d».3]  +  [m.4]  +  [>.5]  ; 

and  this  equation  must  be  satisfied  by  the  values  of  #,  y,  z,  &c.  found 
from  (74). 

152.  EXAMPLE.  —  The  arrangement  of  the  calculation  in  the  case 
of  any  other  number  of  unknown  quantities  is  precisely  similar;  and 
to  illustrate  the  entire  process  let  us  take  the  following  equations, 
each  of  which  is  already  multiplied  by  the  square  root  of  its  weight:— 


0.707z  +  2.052y  —  2.3723  —  0.221w  +  6".58  =  0, 
0.471z  +  1.347y  —  1.715s  —  0.085u  +  1  .63  =  0, 
0.260*  +  0.770y  —  0.356z  -f  0.483w  —  4  .40  =  0, 
0.092*  +  0.343?/  -f  0.2352  -f  0.469w  —  10  .21  =  0, 
0.414*  -f  1.204y  —  1.5063  —  0.205w  +  3  .99  =  0, 
0.040*  -f  0.150#  +  0.1042  +  0.206w  —  4  .34  =  0. 

First,  we  derive 

[nn]  =  204.313, 

[o»l  =  +   4.815,  [oa]^  +  0.971, 

[6n]  =  +  12.961,  [06]  =  +  2.821,  [66]  =  +  8.208, 

[en]    -  —  25.697,  [oc]=  —  3.175,  [6c]  =  —  9.168,  [cc]  =  +  11.028, 

[dn]=    -10.218,  [od]  =  —  0.104,  [W]  =  —  0.261,  [cd]  =  +  0.938,  [dd]  =  +  0.594, 

[w»]  =-18.139,  [as]  =  +  0.513,  [6s]  =  +  1.610,  [c»]  =  —  0.377,  [as]  =+1.177. 

The  values  of  [sn],  [as],  [6s],  [cs],  and  [cfe],  found  by  taking  the 
sums  of  the  normal  coefficients,  agree  exactly  with  the  values  com- 
puted directly,  thus  proving  the  calculation  of  these  coefficients. 
The  normal  equations  are,  therefore, 


NUMERICAL   EXAMPLE.  419 

0.971*  -f  2.821y  —   3.1752  —  0.104w  -f    4.815  =  0, 

2.821ar  -f-  8.208y  —    9.168s  —  O.'Zdlu  -f  12.961  =  0, 

—  3.175*  —  9.168y  +  11.0282  -f-  0.938it  —  25.697  =  0, 

-  0.104*  —  0.251y  +    0.9382  -f-  0.594u  —  10.218  =  0. 

It  will  be  observed  that  the  coefficients  in  these  equations  are  nu- 
merically greater  than  in  the  equations  of  condition;  and  this  will 
generally  be  the  case.  Hence,  if  we  use  logarithms  of  five  decimals 
in  forming  the  normal  equations,  it  will  be  expedient  to  use  tables 
of  six  or  seven  decimals  in  the  solution  of  these  equations. 

Arranging  the  process  of  elimination  in  the  most  convenient  form, 
the  successive  results  are  as  follows : — 


=  +  0.0123,        [fcc.l]  =  +  0.0562,        [6*1]  =  +  0.0511,        [bs.l]  =  +  0.1196,  [fenl]  =  —   1.0278, 

[cc.l]  =  -f  0.6463,        [ccZ.l]  =  +  0.5979,        [cs.l]  =  +  1.3004,  [cn.l]  =  —   9.9528, 

[cc.2]  =  +  0.3895,        [cd.2]  =  +  0.3644,        [cs.2]  =  +  0.7539,  [cn.2]  =  -   5.2567, 

[dd.l]  =  +  0.5829,        [ds.l]  =  +  1.2319,  [dn.l]  =  —  9.7023, 

[dd.2]  =  +  0.3706,        [ds.2]  =  +  0.7350,  [dre.2]  =  —   5.4323, 

[cW.3]  =  +  0.0297,        [ds.3]  =  -f-  0.0297  [dn.3]  =  —  0.5143, 

[nw.l]  =  180.436,  [sn.l]  =  —  20.6828, 

[nn.2]=   94.552,  [sn.2]  =  — 10.6889, 

[nn.3]=   23.608,  [«i.3]=—   0.5143, 

[wi.4]=   14.698,  [sn.4]  =  0. 

The  several  checks  agree  completely,  and  only  the  value  of  [rw.4] 
remains  to  be  proved.     The  equations  (74)  therefore  give 

*  -f  2.9052y  —  3.26982  —  0.1071w  +    4.9588  ==  0, 

y  +  4.56912  +  4.1545w  —  83.5610  =  0, 

z  +  0.9356w  —  13.4960  =  0, 

u  — 17.3165  =  0, 

and  from  these  we  get 

u  =  -f  17".316,       z  =  —  2".705,       y  =  +  23".977,       *  =  —  81".608. 
Then  the  equation  (135)  becomes 

0  r=  +  0.9710*  +  2.8333y  —  2.72932  +  0.3412w  —  1.9838, 

which  is  satisfied  by  the  preceding  values  of  the  unknown  quantities. 
If  we  substitute  these  values  of  x,  y,  z,  and  u  in  the  equations  of 
condition  already  reduced  to  the  same  weight  by  multiplication  by 
the  square  roots  of  their  weights,  we  obtain  the  residuals 

i  Q"  gj        -^f  g^          i   2"  17        2"  01        0".40        0".72 

The  sum  of  the  squares  of  these  gives 

[w]  =  {nnA']  =  11.672, 
and  the  difference  between  this  result  and  the  value  14.698  already 


420  THEORETICAL   ASTRONOMY. 

found  is  due  to  the  decimals  neglected  in  the  computation  of  the 
numerical  values  of  the  several  auxiliaries.  The  sum  of  all  the 
equations  of  condition  gives  generally 

to*  +  l^y  +  M*  +  ld]u  +  ....+  [n]  -  0],          (136) 

which  may  be  used  to  check  the  substitution  of  the  numerical  values 
in  the  determination  of  v,  v'9  &c.  Thus,  we  have,  for  the  values 
here  given, 

1.984s  +  5.866y  —  5.610z  +  0.647w  —  6.75  =  [>]  =  —  l."63. 

It  remains  yet  to  determine  the  relative  weights  of  the  resulting 
values  of  the  unknown  quantities.  For  this  purpose  we  may  apply 
any  of  the  various  methods  already  given.  The  weights  of  u  and  z 
may  be  found  directly  from  the  auxiliaries  whose  values  have  been 
computed.  Thus,  we  have 


p.  =  [<H.3]  =  0.0297,  p,  =  -         [cc.2]  =  0.0312. 


If  we  now  completely  reverse  the  order  of  elimination  from  the 
normal  equations,  and  determine  x  first,  we  obtain  the  values 

[66.2]  =  +  0.0425,  [oa.2]  =  +  0.0033, 

[aa.3]  =  +  0.00056,  [nn.4]  —  14.665, 

and  also 

x  r~  —  82/750,       y  =  +  24."365,       z  =  —  2."699,      u  =  +  17.  "272. 

The  small  differences  between  these  results  and  those  obtained  by  the 
first  elimination  arise  from  the  decimals  neglected.  This  second 
elimination  furnishes  at  once  the  weights  of  x  and  y,  namely, 

P,  =  l>*-3]  —  0.00056,         p9  =  |^||  [66.2]  —  0.0072. 

We  may  also  compute  the  weights  by  means  of  the  equations  (96). 
Thus,  to  find  the  weight  of  y,  we  have 


[cW.21  =  [cM.l]  -  -  [crf.l]  =  +  0.02977, 

and  hence 


The  equations  (103)  and  (108)  are  convenient  for  the  determination 
of  the  values  and  weights  of  the  unknown   quantities   separately. 


CORRECTION   OF   THE   ELEMENTS.  421 

Thus,  by  means  of  the  values  of  the  auxiliaries  obtained  in  the  first 
elimination,  we  find  from  the  equations  (100),  (101),  and  (102), 

A'  =.  —  2.9052,  A"  =  -f  16.5442,  A'"  =  —  3.3012, 

#'  =  —  4.5691,  JB"'  =  +    0.1202,  C""  =  —  0.9356, 

and  then  the  equations  (103)  and  (108)  give 

e  =  —  81".609,      y  =  +  23".977,       z  =  —  2".705,      u  =--  +  17".316f 
Pm  =  0.00057,         pv  =  0.0074,  A  =  0.0312,        Pv  =  0.0297, 

agreeing  with  the  results  obtained  by  means  of  the  other  methods. 
The  weights  are  so  small  that  it  may  be  inferred  at  once  that  the 
values  of  x,  y,  z,  and  u  are  very  uncertain,  although  they  are  those 
which  best  satisfy  the  given  equations.  It  will  be  observed  that  if 
we  multiply  the  first  normal  equation  by  2.9,  the  resulting  equation 
will  differ  very  little  from  the  second  normal  equation,  and  hence  we 
have  nearly  the  case  presented  in  which  the  number  of  independent 
relations  is  one  less  than  the  number  of  unknown  quantities. 

The  uncertainty  of  the  solution  will  be  further  indicated  by  deter- 
mining the  probable  errors  of  the  results,  although  on  account  of  the 
small  number  of  equations  the  probable  or  mean  errors  obtained  may 
be  little  more  than  rude  approximations.  Thus,  adopting  the  value 
of  [vv]  obtained  by  direct  substitution,  we  have 


and  hence 

r=±  1".629, 

which  is  the  probable  error  of  the  absolute  term  of  an  equation  of 
condition  whose  weight  is  unity.  Then  the  equations 

r  r  r      0 

r  =     .  —  ,  r  =     .  —  ,  r,  =    ,  —  >  &c., 

*     Vpl          '     VP,          '     Vp. 

give 

rx  =  ±  68".25,         ry  =  ±  18".94,        rz  =  ±  9".22,         ru  =  ±  9".45. 

It  thus  appears  that  the  probable  error  of  z  exceeds  the  value  obtained 
for  the  quantity  itself,  and  that  although  the  sum  of  the  squares  of 
the  residuals  is  reduced  from  204.31  to  11.67,  the  results  are  still 
quite  uncertain. 

153.  The  certainty  of  the  solution  will  be  greatest  when  the  coef- 
ficients in  the  equations  of  condition  and  also  in  the  normal  equation** 


422  THEORETICAL   ASTRONOMY. 

differ  very  considerably  both  in  magnitude  and  in  sign.  In  the  cor- 
rection of  the  elements  of  the  orbit  of  a  planet  when  the  observa- 
tions extend  only  over  a  short  interval  of  time,  the  coefficients  will 
generally  change  value  so  slowly  that  the  equations  for  the  direct 
determination  of  the  corrections  to  be  applied  to  the  elements  will 
not  afford  a  satisfactory  solution.  In  such  cases  it  will  be  expedient 
to  form  the  equations  for  the  determination  of  a  less  number  of 
quantities  from  which  the  corrected  elements  may  be  subsequently 
derived.  Thus  we  may  determine  the  corrections  to  be  applied  to 
two  assumed  geocentric  distances  or  to  any  other  quantities  which 
afford  the  required  convenience  in  the  solution  of  the  problem, 
various  formulae  for  which  have  been  given  in  the  preceding  chapter. 
The  quantities  selected  for  correction  should  be  known  functions  of 
the  elements,  and  such  that  the  equations  to  be  solved,  in  order  to 
combine  all  the  observed  places,  shall  not  be  subject  to  any  uncer- 
tainty in  the  solution.  But  when  the  observations  extend  over  a  long 
period,  the  most  complete  determination  of  the  corrections  to  be 
applied  to  the  provisional  elements  will  be  obtained  by  forming  the 
equations  for  these  variations  directly,  and  combining  them  as  already 
explained.  A  complete  proof  of  the  accuracy  of  the  entire  calcula- 
tion will  be  obtained  by  computing  the  normal  places  directly  from 
the  elements  as  finally  corrected,  and  comparing  the  residuals  thus 
derived  with  those  given  by  the  substitution  of  the  adopted  values 
of  the  unknown  quantities  in  the  original  equations  of  condition. 

If  the  elements  to  be  corrected  differ  so  much  from  the  true  values 
that  the  squares  and  products  of  the  corrections  are  of  sensible  mag- 
nitude, so  that  the  assumption  of  a  linear  form  for  the  equations  does 
not  afford  the  required  accuracy,  it  will  be  necessary  to  solve  the 
equations  first  provisionally,  and,  having  applied  the  resulting  cor- 
rections to  the  elements,  we  compute  the  places  of  the  body  directly 
from  the  corrected  elements,  and  the  differences  between  these  and 
the  observed  places  furnish  new  values  of  n,  nf,  n",  &c.,  to  be  used 
in  a  repetition  of  the  solution.  The  corrections  which  result  from 
the  second  solution  will  be  small,  and,  being  applied  to  the  elements 
as  corrected  by  the  first  solution,  will  furnish  satisfactory  results.  In 
this  new  solution  it  will  not  in  general  be  necessary  to  recompute  the 
coefficients  of  the  unknown  quantities  in  the  equations  of  condition, 
since  the  variations  of  the  elements  will  not  be  large  enough  to  affect 
sensibly  the  values  of  their  differential  coefficients  with  respect  to 
the  observed  spherical  co-ordinates.  Cases  may  occur,  however,  in 
vhich  it  may  become  necessary  to  recompute  the  coefficients  of  one 


CORRECTION   OF   THE   ELEMENTS.  423 

or  more  of  the  unknown  quantities,  but  only  when  these  coefficients 
are  very  considerably  changed  by  a  small  variation  in  the  adopted 
values  of  the  elements  employed  in  the  calculation.  In  such  cases 
the  residuals  obtained  by  substitution  in  the  equations  of  condition 
will  not  agree  with  those  obtained  by  direct  calculation  unless  the 
corrections  applied  to  the  corresponding  elements  are  very  small.  It 
may  also  be  remarked  that  often,  and  especially  in  a  repetition  of  the 
solution  so  as  to  include  terms  of  the  second  order,  it  will  be  suffi- 
ciently accurate  to  relax  a  little  the  rigorous  requirements  of  a  com- 
plete solution,  and  use,  instead  of  the  actual  coefficients,  equivalent 
numbers  which  are  more  convenient  in  the  numerical  operations  re- 
quired. Although  the  greatest  confidence  should  be  placed  in  the 
accuracy  of  the  results  obtained  as  far  as  possible  in  strict  accordance 
with  the  requirements  of  the  theory,  yet  the  uncertainty  of  the  deter- 
mination of  the  relative  weights  in  the  combination  of  a  series  of 
observations,  as  well  as  the  effect  of  uneliminated  constant  errors, 
may  at  least  warrant  a  little  latitude  in  the  numerical  application, 
provided  that  the  weights  of  the  results  are  not  thereby  much  affected. 
A  constant  error  may  in  fact  be  regarded  as  an  unknown  quantity  to 
be  determined,  and  since  the  effect  of  the  omission  of  one  of  the 
unknown  quantities  is  to  diminish  the  probable  errors  of  the  resulting 
values  of  the  others,  it  is  evident  that,  on  account  of  the  existence  of 
constant  errors  not  determined,  the  values  of  the  variables  obtained 
by  the  method  of  least  squares  from  different  corresponding  series  of 
observations  may  differ  beyond  the  limits  which  the  probable  errors 
of  the  different  determinations  have  assigned.  Further,  it  should  be 
observed  that,  on  account  of  the  unavoidable  uncertainty  in  the  esti- 
mation of  the  weights  of  the  observations  in  the  preliminary  combi- 
nation, the  probable  error  of  an  observed  place  whose  weight  is 
unity  as  determined  by  the  final  residuals  given  by  the  equations  of 
condition,  may  not  agree  exactly  with  that  indicated  by  the  prior 
discussion  of  the  observations. 

154.  In  the  case  of  very  eccentric  orbits  in  which  the  corrections 
to  be  applied  to  certain  elements  are  not  indicated  with  certainty  by 
the  observations,  it  will  often  become  necessary  to  make  that  whose 
weight  is  very  small  the  last  in  the  elimination,  and  determine  the 
other  corrections  as  functions  of  this  one;  and  whenever  the  coeffi- 
cients of  two  of  the  unknown  quantities  are  nearly  equal  or  have 
nearly  the  same  ratio  to  each  other  in  all  the  different  equations  of 
condition,  this  method  is  indispensable  unless  the  difficulty  is  reme 


424  THEORETICAL   ASTRONOMY. 

died  by  other  means,  such  as  the  introduction  of  different  elements  or 
different  combinations  of  the  same  elements.  The  equations  (113) 
furnish  the  values  of  the  unknown  quantities  when  we  neglect  that 
which  is  to  be  determined  independently;  and  then  the  equations 
(114)  give  the  required  expressions  for  the  complete  values  of  these 
quantities.  Thus,  when  a  comet  has  been  observed  only  during  a 
brief  period,  the  ellipticity  of  the  orbit,  however,  being  plainly  indi- 
cated by  the  observations,  the  determination  of  the  correction  to  be 
applied  to  the  mean  daily  motion  as  given  by  the  provisional  ele- 
ments, in  connection  with  the  corrections  of  the  other  elements,  will 
necessarily  be  quite  uncertain,  and  this  uncertainty  may  very  greatly 
affect  all  the  results.  Hence  the  elimination  will  be  so  arranged  that 
A//  shall  be  the  last,  and  the  other  corrections  will  be  determined  as 
functions  of  this  quantity.  The  substitution  of  the  results  thus 
derived  in  the  equations  of  condition  will  give  for  each  residual  an 
expression  of  the  following  form  :  — 


Therefore  we  shall  have 

[>]  =  [Vo]  +  2  [V]  AM  -f 
which  may  be  applied  more  conveniently  in  the  equivalent  form 

M  =  [Vo]  -     d  [v]  +  M    AM  +  [         '.  (138) 


The  most  probable  value  of  A/Z  will  be  that  which  renders  [vv]  a 
minimum,  or 

(139) 


and   the   corresponding   value  of  the   sum  of  the   squares  of  the 
residuals  is 

l-  (140) 


The  correction  given  by  equation  (139)  having  been  applied  to  /*, 
the  result  may  be  regarded  as  the  most  probable  value  of  that  ele- 
ment, and  the  corresponding  values  of  the  corrections  of  the  other 
elements  as  determined  by  the  equations  (114)  having  been  also  duly 
applied,  we  obtain  the  most  probable  system  of  elements.  These, 
however,  may  still  be  expressed  in  the  form 


-f  QAM,  &c. 


CORRECTION   OF  THE   ELEMENTS.  425 

the  coefficients  AQJ  Bw  C0,  &c.  being  those  given  by  the  equations 
(114),  and  thus  the  elements  may  be  derived  which  correspond  to  any 
assumed  value  of  p  differing  from  its  most  probable  value.  The 
unknown  quantity  A/*  will  also  be  retained  in  the  values  of  the 
residuals.  Hence,  if  we  assign  small  increments  to  //,  it  nrjy  easily 
be  seen  how  much  this  element  may  differ  from  its  mo^fi  probable 
value  without  giving  results  for  the  residuals  which  are  i/v,ompatible 
with  the  evidence  furnished  by  the  observations. 

If  the  dimensions  of  the  orbit  are  expressed  by  mearx.*  of  the  ele- 
ments q  and  e,  it  may  occur  that  the  latter  will  not  be  determined 
with  certainty  by  the  observations,  and  hence  it  should  be  treated  as 
suggested  in  the  case  of  //;  and  we  proceed  in  a  similar  manner  when 
the  correction  to  be  applied  to  a  given  value  of  tL;  semi-transverse 
axis  a  is  one  of  the  unknown  quantities  to  be  dete  mined. 


426  THEORETICAL   ASTRONOMY, 


CHAPTER  VIII. 

INVESTIGATION  OP  VARIOUS   FORMULAE  FOR  THE  DETERMINATION  OF  THE  SPECIAL 
PERTURBATIONS  OF   A   HEAVENLY  BODY. 

155.  WE  have  thus  far  considered  the  circumstances  of  the  undis- 
turbed motion  of  the  heavenly  bodies  in  their  orbits;  but  a  complete 
determination  of  the  elements  of  the  orbit  of  any  body  revolving 
around  the  sun,  requires  that  we  should  determine  the  alterations  in 
its  motion  due  to  the  action  of  the  other  bodies  of  the  system.  For 
this  purpose,  we  shall  resume  the  general  equations  (18)1?  namely, 


m)   , 


which  determine  the  motion  of  a  heavenly  body  relative  to  the  sun 
when  subject  to  the  action  of  the  other  bodies  of  the  system.  We 
have,  further, 

m'    /I       xaf  +  yy  +  «t  \    ,      m"    /  1       xx"+  yf+  zz"  \ 
'-lm\  r'3  J^lm\'  r"*  P 


which  is  called  the  perturbing  function,  of  which  the  partial  differen- 
tial coefficients,  with  respect  to  the  co-ordinates,  are 


dQ_      m'    Ix'-x      ^\         m"    lx"-x       x" 
fa-l+m\     ff          r'*l+l  +  m\     p"          r"> 


dQ  _      m'    [z'  —  z       J_\    ,      m"    jz"  —  z       4'  \    ,    ^ 
•S"~rfm\     P3          r^/^l+ml     ?'*         r"*  /  " 


and  in  which  m',  mr/,  &c.  denote  the  ratios  of  the  masses  of  the 
several  disturbing  planets  to  the  mass  of  the  sun,  and  m  the  ratio  of 
the  mass  of  the  disturbed  planet  to  that  of  the  sun.  These  partial 
differential  coefficients,  when  multiplied  by  &2(l-f-w),  express  thn 


PERTURBATIONS.  427 

sum  of  the  components  of  the  disturbing  force  resolved  in  directions 
parallel  to  the  three  rectangular  axes  respectively. 

When  we  neglect  the  consideration  of  the  perturbations,  the  general 
equations  of  motion  become 


dt* 

/7/2         '  ^         '  J    „,,  3  '  \*^y 


the  complete  integration  of  which  furnishes  as  arbitrary  constants  of 
integration  the  six  elements  which  determine  the  orbitual  motion  of  a 
heavenly  body.  But  if  we  regard  these  elements  as  representing  the 
actual  orbit  of  the  body  for  a  given  instant  of  time  t,  and  conceive 
of  the  effect  of  the  disturbing  forces  due  to  the  action  of  the  other 
bodies  of  the  system,  it  is  evident  that,  on  account  of  the  change 
arising  from  the  force  thus  introduced,  the  body  at  another  instant 
different  from  the  first  will  be  moving  in  an  orbit  for  which  the 
elements  are  in  some  degree  different  from  those  which  satisfy  the 
original  equations.  Although  the  action  of  the  disturbing  force  is 
continuous,  we  may  yet  regard  the  elements  as  unchanged  during  the 
element  of  time  dt,  and  as  varying  only  after  each  interval  dt.  Let 
us  now  designate  by  t0  the  epoch  to  which  the  elements  of  the  orbit 
belong,  and  let  these  elements  be  designated  by  Mw  TTO,  &0,  iw  e0,  and 
a0;  then  will  the  equations  (3)  be  exactly  satisfied  by  means  of  the 
expressions  for  the  co-ordinates  in  terms  of  these  rigorously-constant 
elements.  These  elements  will  express  the  motion  of  the  body  sub- 
ject to  the  action  of  the  disturbing  forces  only  during  the  infinitesimal 
interval  dt,  and  at  the  time  tQ  +  dt  it  will  commence  to  describe  a 
new  orbit  of  which  the  elements  will  differ  from  these  constant  ele- 
ments by  increments  which  are  called  the  perturbations. 

According  to  the  principle  of  the  variation  of  parameters,  or  of 
the  constants  of  integration,  the  differential  equations  (1)  will  be 
satisfied  by  integrals  of  the  same  form  as  those  obtained  when  the 
second  members  are  put  equal  to  zero,  provided  only  that  the  arbitrary 
constants  of  the  latter  integration  are  no  longer  regarded  as  pure 
constants  but  as  subject  to  variation.  Consequently,  if  we  denote  the 
variable  elements  by  M,  TT,  &,  i,  e,  and  a,  they  will  be  connected 
with  the  constant  elements,  or  those  which  determine  the  orbit  at  the 
instant  £0,  by  the  equations 


428  THEORETICAL   ASTRONOMY. 


di 


(4) 


in  which  —7-,  -r-  .  &c.  denote  the  differential  coefficients  of  the  ele- 
at     at 

ments  depending  on  the  disturbing  forces.  When  these  differential 
coefficients  are  known,  we  may  determine,  by  simple  quadrature,  the 
perturbations  dM,  dn,  &c.  to  be  added  to  the  constant  elements  in 
order  to  obtain  those  corresponding  to  any  instant  for  which  the 
place  of  the  body  is  required.  These  differential  coefficients,  however, 
are  functions  of  the  partial  differential  coefficients  of  Q  with  respect 
to  the  elements,  and  before  the  integration  can  be  performed  it 
becomes  necessary  to  find  the  expressions  for  these  partial  differential 
coefficients.  For  this  purpose  we  expand  the  function  Q  into  a  con- 
verging series  and  then  differentiate  each  term  of  this  series  relatively 
to  the  elements.  This  function  is  usually  developed  into  a  converg- 
ing series  arranged  in  reference  to  the  ascending  powers  of  the  eccen- 
tricities and  inclinations,  and  so  as  to  include  an  indefinite  number 
of  revolutions;  and  the  final  integration  will  then  give  what  are 
called  the  absolute  or  general  perturbations.  When  the  eccentricities 
and  inclinations  are  very  great,  as  in  the  case  of  the  comets,  this 
development  and  analytical  integration,  or  quadrature,  becomes  no 
longer  possible,  and  even  when  it  is  possible  it  may,  on  account  of 
the  magnitude  of  the  eccentricity  or  inclination,  become  so  difficult 
that  we  are  obliged  to  determine,  instead  of  the  absolute  perturbations, 
what  are  called  the  special  perturbations,  by  methods  of  approxima- 
tion known  as  mechanical  quadratures,  according  to  which  we  deter- 
mine the  variations  of  the  elements  from  one  epoch  tQ  to  another 
epoch  t.  This  method  is  applicable  to  any  case,  and  may  be  advan- 
tageously employed  even  when  the  determination  of  the  absolute 
perturbations  is  possible,  and  especially  when  a  series  of  observations 
extending  through  a  period  of  many  years  is  available  and  it  is 
desired  to  determine,  for  any  instant  £0,  a  system  of  elements,  usually 
called  osculating  elements,  on  which  the  complete  theory  of  the  motion 
may  be  based. 

Instead  of  computing  the  variations  of  the  elements  of  the  orbit 
directly,  we  may  find  the  perturbations  of  any  known  functions  of 
these  elements;  and  the  most  direct  and  simple  method  is  to  deter- 
mine the  variations,  due  to  the  action  of  the  disturbing  forces,  of 
any  system  of  three  co-ordinates  by  means  of  which  the  position  of 


PERTURBATIONS.  429 

the  body  or  the  elements  themselves  may  be  found.  We  shall,  there- 
fore, derive  various  formulae  for  this  purpose  before  investigating  th<» 
formulae  for  the  direct  variation  of  the  elements. 

156.  Let  XQ,  y0,  z0  be  the  rectangular  co-ordinates  of  the  body  at 
the  time  t  computed  by  means  of  the  osculating  elements  Mw  TTO,  Q0, 
&c.,  corresponding  to  the  epoch  t0.  Let  x,  y,  z  be  the  actual  co-ordi- 
nates of  the  disturbed  body  at  the  time  t;  and  we  shall  have 


dx,  dy,  and  dz  being  the  perturbations  of  the  rectangular  co-ordinates 
from  the  epoch  t0  to  the  time  t.  If  we  substitute  these  values  of  x, 
y,  and  z  in  the  equations  (1),  and  then  subtract  from  each  the  corre- 
sponding one  of  equations  (3),  we  get 


Let  us  now  put  r  =  r0  -f-  dr;  then  to  terms  of  the  order  ^r2,  which  is 
equivalent  to  considering  only  the  first  power  of  the  disturbing  force, 
we  have 


and  hence 


We  have  also  from 

r'-z'  +  ^ 
neglecting  terms  of  the  second  order, 


(7) 


430  THEORETICAL   ASTRONOMY. 

The  integration  of  the  equations  (6)  will  give  the  perturbations  dx, 
dy,  and  dz  to  be  applied  to  the  rectangular  co-ordinates  xw  yw  z0  com- 
puted by  means  of  the  osculating  elements,  in  order  to  find  the  actual 
co-ordinates  of  the  body  for  the  date  to  which  the  integration  belongs. 
But  since  the  second  members  contain  the  quantities  dx9  %,  dz  which 
are  sought,  the  integration  must  be  effected  indirectly  by  successive 
approximations;  and  from  the  manner  in  which  these  are  involved 
in  the  second  members  of  the  equations,  it  will  appear  that  this  inte- 
gration is  possible. 

If  we  consider  only  a  single  disturbing  planet,  according  to  the 
equations  (2),  we  shall  have 


*'  \ 

7')' 


and  these  forces  we  will  designate  by  X,  Y,  and  Z  respectively  ;  then, 
if  in  these  expressions  we  neglect  the  terms  of  the  order  of  the 
square  of  the  disturbing  force,  writing  .r0,  y0,  ZQ  in  place  of  x,  y,  z, 
the  equations  (6)  become 


df  r,» 

(9) 


which  are  the  equations  for  computing  the  perturbations  of  the  rec- 
tangular co-ordinates  with  reference  only  to  the  first  power  of  the 
masses  or  disturbing  forces.  We  have,  further, 

,«  =  (gf  _  .,.)«  +  ft  -  y)«  -I-  (/  _  zy,  (10) 

in  which,  when  terms  of  the  second  order  are  neglected,  we  use  the 
values  xQ,  y0,  z0  for  x,  y,  and  z  respectively. 

1 57.  From  the  values  of  8x,  dy,  and  dz  computed  with  regard  to 
the  first  power  of  the  masses  we  may,  by  a  repetition  of  part  of  the 
calculation,  take  into  account  the  squares  and  products  and  even  the 
higher  powers  of  the  disturbing  forces.  The  equations  (5)  may  be 
written  thus: — 


VAEIATION   OF   CO-OKDINATES.  431 


d? 


in  which  nothing  is  neglected.  In  the  application  of  these  formulae, 
as  soon  as  dx,  dy,  and  $2  have  been  found  for  a  few  successive  inter- 
vals, we  may  readily  derive  approximate  values  of  these  quantities 
for  the  date  next  following,  and  with  these  find 

x  =  x0-\-dx,  y  =  2/0  +  dy,  Z  =  ZQ-\-  dz, 

and  hence  the  complete  values  of  the  forces  X,  Y,  and  Z,  by  means 
of  the  equations  (8).  To  find  an  expression  for  the  factor 

«-5f 

which  will  be  convenient  in  the  numerical  calculation,  we  have 


and  therefore 


.  =  x       2  g0  *      (y0 

2  ~T~ 


Let  us  now  put 

?= 

and 

/}=i 

then  we  shall  have 


and  the  values  of/  may  be  tabulated  with  the  argument  q.     The 
equations  (11)  therefore  become 


(14) 


432  THEOKETICAL   ASTEONOMY. 

The  coefficients  of  8x,  dy,  and  dz  in  equation  (12)  may  be  found  at 
once,  with  sufficient  accuracy,  by  means  of  the  approximate  values 
of  these  quantities;  and  having  found  the  value  of  / corresponding 

to  the  resulting  value  of  g,  the  numerical  values  of     ,  2  >      ,  f  >  and 

d*8z 

-ITT,  which  include  the  squares  and  products  of  the  masses,  will  be 

obtained.  The  integration  of  these  will  give  more  exact  values  of 
dx,  dy,  and  dz,  and  then,  recomputing  q  and  the  other  quantities  which 
require  correction,  a  still  closer  approximation  to  the  exact  values  of 
the  perturbations  will  result. 

Table  XVII.  gives  the  values  of  log/  for  positive  or  negative 
values  of  q  at  intervals  of  0.0001  from  q  =  0  to  q  =  0.03.  Unless 
the  perturbations  are  very  large,  q  will  be  found  within  the  limits  of 
this  table;  and  in  those  cases  in  which  it  exceeds  the  limits  of  the 
table,  the  value  of 


may  be  computed  directly,  using  the  value  of  r  in  terms  of  rQ  and 
dx,  dy,  dz. 

In  the  application  of  the  preceding  formulae,  the  positions  of  the 
disturbed  and  disturbing  bodies  may  be  referred  to  any  system  of 
rectangular  co-ordinates.  It  will  be  advisable,  however,  to  adopt 
either  the  plane  of  the  equator  or  that  of  the  ecliptic  as  the  funda- 
mental plane,  the  positive  axis  of  x  being  directed  to  the  vernal 
equinox.  By  choosing  the  plane  of  the  elliptic  orbit  at  the  time  t0 
as  the  plane  of  xy,  the  co-ordinate  z  will  be  of  the  order  of  the  per- 
turbations, and  the  calculation  of  this  part  of  the  action  of  the  dis- 
turbing force  will  be  very  much  abbreviated;  but  unless  the  inclina- 
tion is  very  large  there  will  be  no  actual  advantage  in  this  selection, 
since  the  computation  of  the  values  of  the  components  of  the  dis- 
turbing forces  will  require  more  labor  than  when  either  the  equator 
or  the  ecliptic  is  taken  as  the  fundamental  plane.  The  perturbations 
computed  for  one  fundamental  plane  may  be  converted  into  those 
referred  to  another  plane  or  to  a  different  position  of  the  axes  in  the 
same  plane  by  means  of  the  formulae  which  give  the  transformation 
of  the  co-ordinates  directly. 

158.  We  shall  now  investigate  the  formulae  for  the  integration  of 
the  linear  differential  equations  of  the  second  order  which  express  the 
variation  of  the  co-ordinates,  and  generally  the  formulae  for  finding 

the  integrals  of  expressions  of  the  form  J  f(x)dx  and  JJ  J(x)dx* 


MECHANICAL   QUADRATURE.  433 

when  the  values  of  f(x)  are  computed  for  successive  values  of  x  in- 
creasing in  arithmetical  progression.  First,  therefore,  we  shall  find 
the  integral  of  f(x)  dx  within  given  limits. 

Within  the  limits  for  which  x  is  continuous,  we  have 


f(x)  =  a  +  {3x  +  rx*  +  W+ex*+....;  (15) 

and  if  we  consider  only  three  terms  of  this  series,  the  resulting  equa- 
tion 

f(x)  ^a  +  px  +  rx* 

is  that  of  the  common  parabola  of  which  the  abscissa  is  x  and  the 
ordinate  /(#),  and  the  integral  of  f(x)  dx  is  the  area  included  by  the 
abscissa,  two  ordinates,  and  the  included  arc  of  this  curve.  Gene- 
rally, therefore,  we  may  consider  the  more  complete  expression  for 
f(x)  as  the  equation  of  a  parabolic  curve  whose  degree  is  one  less 
than  the  number  of  terms  taken.  Hence,  if  we  take  n  terms  of  the 
series  as  the  value  of/(#),  we  shall  derive  the  equation  for  a  parabola 
whose  degree  is  n  —  1,  and  which  has  n  points  in  common  with  the 
curve  represented  by  the  exact  value  of  f(x). 

If  we  multiply  equation  (15)  by  dx  and  integrate  between  the 
limits  0  and  x',  we  get 


dx  = 


If  now  the  values  of  f(x)  for  different  values  of  x  from  0  to  xf  are 
known,  each  of  these,  by  means  of  equation  (15),  will  furnish  an 
equation  for  the  determination  of  a,  /?,  7-,  &c.  ;  and  the  number  of 
terms  which  may  be  taken  will  be  equal  to  the  number  of  different 
known  values  of  /(a?).  As  soon  as  a,  /?,  7-,  &c.  have  thus  been  found, 
the  equation  (16)  will  give  the  integral  required. 

If  the  values  of  f(x)  are  computed  for  values  of  x  at  equal  inter- 
vals and  we  integrate  between  the  limits  x  =  0,  and  x  =  n&x,  &x 
being  the  constant  interval  between  the  successive  values  of  xy  and 
n  the  number  of  intervals  from  the  beginning  of  the  integration,  we 
obtain 


x 

Cf(x)  dx  = 


Let  us  now  suppose  a  quadratic  parabola  to  pass  through  the  points 
of  the  curve  represented  by  f(x),  corresponding  to  x  =  0,  x  =  &x, 


28 


434  THEORETICAL   ASTRONOMY. 

and  x  =  2&x;  then  will  the  area  included  by  the  arc  of  this  parabola, 
the  extreme  ordinates,  and  the  axis  of  abscissas  be 


'Ibx 

J/(aO'<fe  =  A*(2< 


The  equation  of  the  curve  gives,  if  we  designate  the  ordinates  of  the 
three  successive  points  by  yQ,  yly  and  y2, 

»  =  2/o,        P  =  — 
and  hence  we  derive 

2Aa? 


In  a  similar  manner,  the  area  included  by  the  ordinates  y2  and  y4, — 
corresponding  to  a;  =  2A#  and  a?  =  4&x, — the  axis  of  abscissas,  and 
the  parabola  passing  through  the  three  points  corresponding  to  y2,  y# 
and  y4,  is  found  to  be 


2Aa; 

and  hence  we  have,  finally, 

nAx 

J/0)  ^  = 

(n  —  2)  A* 

The  sum  of  all  these  gives 


(17) 


by  means  of  which  the  approximate  value  of  the  integral  within  the 
given  limits  may  be  found. 

If  we  consider  the  curve  which  passes  through  four  points  corre- 
sponding to  yw  ylf  ya,  and  yB9  we  have 


for  the  equation  of  the  curve,  and  hence,  giving  to  x  the  values  0, 
;,  and  SAO?,  successively,  we  easily  find 


MECHANICAL   QUADRATURE.  435 


«  =  yo» 
i 


Therefore  we  shall  have 


In  like  manner,  by  taking  successively  an  additional  term  of  the 
series,  we  may  derive 


*)  dx  =         (7y.  +  32yt  +  12yf  +  32y. 

x  (19) 

=          (19y0  +  76y,  +  50y3  +  50y3 


This  process  may  be  continued  so  as  to  include  the  extreme  values  of 
x  for  which  /(a?)  is  known;  but  in  the  calculation  of  perturbations  it 
will  be  more  convenient  to  use  the  finite  differences  of  the  function 
instead  of  the  function  itself  directly.  We  may  remark,  further, 
that  the  intervals  of  quadrature  when  the  function  itself  is  used, 
may  be  so  determined  that  the  degree  of  approximation  will  be  much 
greater  than  when  these  intervals  are  uniform. 

159.  Let  us  put  &x  =  to,  and  let  the  value  of  x  for  which  n  —  0 
be  designated  by  a;  then  will  the  general  value  be 

/(*)  -/(a  +  n«0, 

a?  being  the  constant  interval  at  which  the  values  of  /(a?)  are  given. 
Hence  we  shall  have 

dx  =  a>dn, 
I  f(x)  dx  =  (o  I  /(a  -j-  TWO)  dn. 


If  we  expand  the  function  /(a  +  no)),  we  have 


na> 


f     . 

.,  (20) 


436  THEOKET1CAL   ASTRONOMY. 

and  hence 

//(a  +  n»)  dn  =  C  +  n/(a)  +  in-  ^  +  JnV 


C  being  the  constant  of  integration.     The  equations  (54)6  give 

*  ^    =/(«)  -  ir  w  +  «w(«)  -  T 


-'        =/"(«)  -  A/*(«)  +  */"(«)  -  »*«/•"(«)  +  •  •  •  . 

=/"  (a)  -  \r  («)  +  T!*/'u  («)-..., 

(22) 

=/"(a)  -  */VI(a)  +  «'/""  (a)  ~  •  •  •  • 

=/'  («)  -if'"  («)+.-., 


iii  which  the  functional  symbols  in  the  second  members  denote  the 
diiferent  orders  of  finite  differences  of  the  function.    Hence  we  obtain 

+  no*)  cfo  =  0  +  nf(a) 

+  in1  (/(a)  -  I/"  (a)  +  A/*  (a)  -  T^/vU(a)  +  .  .  .) 
+  Jn«(/"(a)  -  A/1'  (a)  +  ^/"(a)  -  vhf^(a)  +  ...) 


Ef  we  take  the  integral  between  the  limits  —  nr  and  +n',  the  terms 
containing  the  even  powers  of  n  disappear.  Further,  sinc«  the  values 
of  the  function  are  supposed  to  be  known  for  a  series  of  values  of  n 
at  intervals  of  a  unit,  it  will  evidently  be  convenient  to  determine 
the  integral  between  the  required  limits  by  means  of  the  sum  of  a 
series  of  integrals  whose  limits  are  successively  increased  by  a  unit, 
such  that  the  difference  between  the  superior  and  the  inferior  limit 
of  each  integral  shall  be  a  unit.  Hence  we  take  the  first  integral 
between  the  limits  —  J  and  +J,  and  the  equation  (23)  gives,  after 
reduction, 


MECHANICAL   QUADRATURE.  43? 

-f  * 


f/(a  +  n«0  dn  -/(a)  +  £/"(«)  ~  s4Is/iv(«)  + 
J-i  (24) 

-4gf!fl!^/TiilW  +  &c. 

It  is  evident  that  by  writing,  in  succession,  a  -\-  a),  a  +  2o>,  .... 
a  -f-  ia>  in  place  of  a,  we  simply  add  1  to  each  limit  successively,  so 
that  we  have 

%  <  +  1  +  * 

J/(a  +  na»)  cfo  ==(/((«  +  »«0  +  (»  —  i)  «)  d  (n  —  t) 


But  since 

1  *+* 


-*  -i  * 

if  we  give  to  i  successively  the  values  0,  1,  2,  3,  &c.  in  the  preceding 
equation,  and  add  the  results,  we  get 

i  -f  i  n  =  i  n  =  i 

J/(a  +  n«>)  dn  =  ^f(a  +  n«0  -f-  ^?  ^/'  (a  +  na>) 


t  n  =  i 

rCa  +  ««)  -I-  ^W5^/vl(«  +  n«)  -  Ac. 

n=0  n=0 

Let  us  now  consider  the  functions  /(a),  /(a  +  not),  &c.  as  being 
themselves  the  finite  differences  of  other  functions  symbolized  by  '/, 
the  first  of  which  is  entirely  arbitrary,  so  that  we  may  put,  in  accord- 
ance with  the  adopted  notation, 

/O)  =  '/(«  -f  M  -  '/(»  -  J«0, 
/(a  +  «)  =  '/(a  +  j«,)  -  '/(a  +  J«), 

/(a  +  tia,)  =  '/(a  +  (n  +  J»  —  '/(a  +  (»  -  J)  «>). 
Therefore  we  shall  have 

n  =  t 

Y/O  +  no»)  =  '/(a  +  (i  +  i)  «,)  -  '/(a  -  i«), 

n  =  0 

and  also 

n  =  i 

)  =/  (a  +  (t  +  J)  «)  -/  (a  -  ^), 


433  THEORETICAL   ASTRONOMY. 

Further,  since  the  quantity  ff(a  —  \a>)  is  entirely  arbitrary,  we  may 
assign  to  it  a  value  such  that  the  sum  of  all  the  terms  of  the  equation 
which  have  the  argument  a  —  \a>  shall  be  zero,  namely, 


(26) 
Substituting  these  values  in  (25),  it  reduces  to 


I  f(x)  dx  —  to  I  /(a  -|~  nw)  efoi 

a-fc.  -i  (27) 

=  «{'/(«  +  (*  +  i»  +  &/(«  +  (*'  +  J» 


In  the  calculation  of  the  perturbations  of  a  heavenly  body,  the 
dates  for  which  the  values  of  the  function  are  computed  may  be  so 
arranged  that  for  n  =  —  J,  corresponding  to  the  inferior  limit,  the 
integral  shall  be  equal  to  zero,  the  epoch  of  f(a  —  \co)  being  that  of 
the  osculating  elements.  It  will  be  observed  that  the  equation  (26) 
expresses  this  condition,  the  constant  of  integration  being  included 
in  '/(a  —  \to).  If,  instead  of  being  equal  to  zero,  the  integral  has  a 
given  value  when  n  =  —  J,  it  is  evidently  only  necessary  to  add  this 
value  to  ff(a  —  Jo>)  as  given  by  (26). 

160.  The  interval  co  and  the  arguments  of  the  function  may  always 
be  so  taken  that  the  equation  (27)  will  furnish  the  required  integral, 
either  directly  or  by  interpolation  ;  but  it  will  often  be  convenient  to 
integrate  for  other  limits  directly,  thus  avoiding  a  subsequent  inter- 
polation. The  derivation  of  the  required  formulae  of  integration 
may  be  effected  in  a  manner  entirely  analogous  to  that  already  indi- 
cated. Thus,  let  it  be  required  to  find  the  expression  for  the  integra1 
taken  between  the  limits  —  \  and  i. 

The  general  formula  (23)  gives 


J, 


"  (a)- 


and  since,  according  to  the  notation  adopted, 

/  («)  =  1  CT  («  -  i")  +/'(«  +  i-)) 

=  f(«  +  i-)    -iT(«),  (28) 

/"(«)=ro»+i«)  -*/"(«), 

/'(«)  =/'(«  +  i«0     -  if"  («),  &e., 


MECHANICAL   QUADRATURE.  439 

this  becomes 


i 

//(a+n-O  cfo=l/(a)+  J/  (a+  j«0-&r  (a)-,!,/''  («+ 


(29> 


Therefore  we  obtain 

<  +  * 

/(a-j-nw)  dn=^f(a+ic 


Now  we  have 

\  f(a  +  nuj)  dn=  I  /(a  -|-  nai]  dn  —  ( /(a  -f-  na>)  dn ; 
-i  -i 

and  if  we  substitute  the  values  already  found  for  the  terms  in  the 
second  member,  and  also 

/" («  +  fa)  =   f(a  +  (,•  +  !)„)_  f(a  +  (i-l) a,), 
/»(«  +  i.)  =/'" (a  +  (t  +  i) «0  -/"' (<•  +  (t  -  i) •), 

we  get 

a +  i<a  t 

/(«)  c?a;  =  o)  I  /(a  -|"  nw)  dn 

J-*  (32) 


which  is  the  required  integral  between  the  limits  — J  and  i. 

161.  The  methods  of  integration  thus  far  considered  apply  to  the 
cases  in  which  but  a  single  integration  is  required,  and  when  applied 
+o  the  integration  of  the  differential  equations  for  the  variations  of 
the  co-ordinates  on  account  of  the  action  of  disturbing  bodies,  they 

_  ddx  ddy  ddz        .          . 

will  only  give  the  values  or  -,->  -TT->  and  — ;->  and  another  integration 

becomes  necessary  in  order  to  obtain  the  values  of  dx,  dy,  and  dz. 
We  will  therefore  proceed  to  derive  formulae  for  the  determination 
of  the  double  integral  directly. 


440  THEORETICAL   ASTRONOMY. 

For  the  double  integral  j  J  f(x)  dx2  we  have,  since  dx*  =  a)2dn2, 


The  value  of  the  function  designated  by  /(a)  being  so  taken  that 
when  n  =  —  \, 

Cf(a  +  n«0  dn  —  0, 

the  equation  (23)  gives 


Therefore,  the  general  equation  is 

o 
J/(a  -f  n<o~)  dn  =  if  (a  +  nw)  dn  -f-  nf(a) 


-f  i»n"  +  tfn*  +  Ar»«  -f-  Ti^n»  +  Ac. 


the  values  of  a,  /?,  ^,  .  .  .  being  given  by  the  equations  (22).     Multi- 
plying this  by  dn,  and  integrating,  we  get 


Cf  being  the  new  constant  of  integration.     If  we  take  the  integral 
between  the  limits  —  |  and  -j-  J,  we  find 


ff/(*  +  n«)  c/rz,2  =  (/(a  +  na»)  ^n  +  ^ 
-i  ^-i 

From  the  equation  (32)  we  get,  for  i  =  0, 

o 
J/O  +  n.)  dn  =  '/(a)  -  ^f  (a)  +  ^'5/'"  (a)  -  gMil/'  («)  +  &<=•  (33; 


Substituting  this  value,  and  also  the  values  of  a,  7-,  e,  &c.,  —  which 
are  given  by  the  second  members  of  the  equations  (22),  —  in  the  pre- 
ceding equation,  and  reducing,  we  get 


-M 
r(fl 

-i 


MECHANICAL  QUADRATURE.  443 

Hence 

jfif/(«  +  no»)  dri*  =  '/(a  -f  io»)  —  3'?/'  (a  +  it*) 

+  lils/'"  (a  H-  **")  —  iAV«ff/T (a  +  tw)  -f-  Ac. 
and 

»  +  t  n  =  t  n  =  t 

/»/»  -V^f  ^-w 

I      I        /•/  I  \        T        n  ^     T    /  /*/•  I  \  j  ^^    "/»//•  v 

I  j(a  -j-  7io>J  o>i  ;=   ^  J(CL  -\-  nw)  —  2?  ^  /  (a  ~t~  ^w) 

«A/  >^—  *—i 

,17Q  n=\_,  (35) 

-f-&c. 


n=0  »=0 


We  may  evidently  consider  '/(a  —  Ja>),  '/(a  +  Jw),  &c.  ae  the  differ- 
ences of  other  functions,  the  first  of  which  is  arbitrary;  40  that  we 
have 

7  (a)  =  ¥f(a  +  -J«0  +  i'/O  -  i«)  -  i"/(a  +  «)    -  if  («  -  «), 

'/(a  +  *)  =  J'/(a  +  i«)  +  i'/(a  +  ^)  -  J"/(a  +  2«)  -  ^/"  (a), 


-f  i'/(«  +  (^-i)«)  =  i7(a  +  ( 
-r/(a  +  (?l_l)w). 

Therefore 


n  =  0 


Substituting  these  values  in  equation  (35),  and  observing  that 


700  +  7(«  -  «)  =  2"/(«  -  •)  +  '/(«  -  -^)> 

/(a)  +   /(a  -  a,)  =    2/(a)          -  /  (a  -  Jo,), 
/'  W  +/"  («—••)«  2/'  (a)          -/"'  (a  -  i«0,  Ac., 


and  that,  since  "f(a  —  to)  is  arbitrary,  we  may  put 

"/(a  -  «,)  = 


(«)  +  2/"  (a  — ))  -  &o., 


(36) 


442  THEORETICAL   ASTRONOMY. 

the  integral  becomes 

*  +  «  +  *><•>        r/  +  * 
JJ  /(*)  dx*  =  «>2JJ  /(a  +  no,)  dn* 

l)«>)    (37) 


which  is  the  expression  for  the  double  integral  between  the  limits 
—  \  and  i  +  J. 

The  value  of  "/(a  —  to)  given  by  equation  (36)  is  in  accordance 
with  the  supposition  that  for  n  =  —  \  the  double  integral  is  equal  to 
zero,  and  this  condition  is  fulfilled  in  the  calculation  of  the  pertur- 
bations when  the  argument  a  —  \a>  corresponds  to  the  date  for  which 
the  osculating  elements  are  given.  If,  for  n  =  —  J,  neither  the  single 
nor  the  double  integral  is  to  be  taken  equal  to  zero,  it  is  only  neces- 
sary to  add  the  given  value  of  the  single  integral  for  this  argument 
to  the  value  of  '/(a  —  Jai)  given  by  equation  (26),  and  to  add  the 
given  value  of  the  double  integral  for  the  same  argument  to  the  value 
of  "/(a  —  ai)  given  by  (36). 

162.  In  a  similar  manner  we  may  find  the  expressions  for  the 
double  integral  between  other  limits.  Thus,  let  it  be  required  to 
find  the  double  integral  between  the  limits  —  J  and  i. 

Between  the  limits  0  and  £  we  have 


jjf(a 


which  gives 


dn  +  J/(a)  +  &. 

+  ssW  +  1*4*3*  +  Ac. 


GO  (38) 


and  this  again,  by  means  of  (28),  gives 

** 


MECHANICAL   QUADRATURE.  443 

Therefore,  since 


CC  rr  rr 

JJ  /O  -f  no»)  dn*  =JJ  f(a  +  no»)  drc,2  —  JJ  /(a  +  n«0  d»», 

-*  -t  »• 

and 

'/(«  +  (*  +  i)  «0  =  7(«  +  (*  +  1;  «)  —  7(«  +  »'«), 

A*  -K*  +  *>»)==  /(a  +  (t  +  l»  —  /(<*  +  *»), 
/'"  (a  +  (i  +  i)  a,)  =/"  (a  +  (i  +  1)  w)  -f"  (a  -f  »«,),  &c. 

we  shall  have 

a  -f  i  w  ^ 

(^)  da?  =  «*0/(a  +  nai)  (fo8 

*w  -i  (39) 


which  gives  the  required  integral  between  the  limits  —  J  and  i. 

163.  It  will  be  observed  that  the  coefficients  of  the  several  terms 
of  the  formulae  of  integration  converge  rapidly,  and  hence,  by  a 
proper  selection  of  the  interval  at  which  the  values  of  the  function 
are  computed,  it  will  not  be  necessary  to  consider  the  terms  which 
depend  on  the  fourth  and  higher  orders  of  differences,  and  rarely 
those  which  depend  on  the  second  and  third  differences.  The  value 
assigned  to  the  interval  a)  must  be  such  that  we  may  interpolate  with 
certainty,  by  means  of  the  values  computed  directly,  all  values  of  the 
function  intermediate  to  the  extreme  limits  of  the  integration;  and 
hence,  if  the  fourth  and  higher  orders  of  differences  are  sensible,  it 
will  be  necessary  to  extend  the  direct  computation  of  the  values  of 
the  function  beyond  the  limits  which  would  otherwise  be  required, 
in  order  to  obtain  correct  values  of  the  differences  for  the  beginning 
and  end  of  the  integration.  It  will  be  expedient,  therefore,  to  take 
(o  so  small  that  the  fourth  and  higher  differences  may  be  neglected, 
but  not  smaller  than  is  necessary  to  satisfy  this  condition,  since  other- 
wise an  unnecessary  amount  of  labor  would  be  expended  in  the 
direct  computation  of  the  values  of  the  function.  It  is  better,  how- 
ever, to  have  the  interval  at  smaller  than  what  would  appear  to  be 
strictly  required,  in  order  that  there  may  be  no  uncertainty  with 
respect  to  the  accuracy  of  the  integration.  On  account  of  the  rapidity 
with  which  the  higher  orders  of  differences  decrease  as  we  diminish 
to,  a  limit  for  the  magnitude  of  the  adopted  interval  will  speedily  be 
obtained.  The  magnitude  of  the  interval  will  therefore  be  suggested 
by  tne  rapidity  of  the  change  of  value  of  the  function.  In  the  coin- 


444  THEORETICAL  ASTRONOMY. 

ptitation  of  the  perturbations  of  the  group  of  small  planets  between 
Mars  and  Jupiter  we  may  adopt  uniformly  an  interval  of  forty  days; 
but  in  the  determination  of  the  perturbations  of  comets  it  will  evi- 
dently be  necessary  to  adopt  different  intervals  in  different  parts  of 
the  orbit.  When  the  comet  is  in  the  neighborhood  of  its  perihelion, 
and  also  when  it  is  near  a  disturbing  planet,  the  interval  must  neces- 
sarily be  much  smaller  than  when  it  is  in  more  remote  parts  of  its 
orbit  or  farther  from  the  disturbing  body. 

It  will  be  observed,  further,  that  since  the  double  integral  contains 
the  factor  o>2,  if  we  multiply  the  computed  values  of  the  function  by 
o>2,  this  factor  will  be  included  in  all  the  differences  and  sums,  and 
hence  it  will  not  appear  as  a  factor  in  the  formulae  of  integration. 
If,  however,  the  values  of  the  function  are  already  multiplied  by  w2, 
and  only  the  single  integral  is  sought,  the  result  obtained  by  the 
formula  of  integration,  neglecting  the  factor  o>2,  will  be  to  times  the 
actual  integral  required,  and  it  must  be  divided  by  a)  in  order  to 
obtain  the  final  result. 

164.  In  the  computation  of  the  perturbations  of  one  of  the  asteroid 
planets  for  a  period  of  two  or  three  years  it  will  rarely  be  necessary 
to  take  into  account  the  effect  of  the  terms  of  the  second  order  with 
respect  to  the  disturbing  force.  In  this  case  the  numerical  values  of 
the  expressions  for  the  forces  will  be  computed  by  using  the  values 
of  the  co-ordinates  computed  from  the  osculating  elements  for  the 
beginning  of  the  integration,  instead  of  the  actual  disturbed  values 
of  these  co-ordinates  as  required  by  the  formulae  (8).  The  values  of 
the  second  differential  coefficients  of  dx,  %,  and  dz  with  respect  to 
the  time,  will  be  determined  by  means  of  the  equations  (9).  If  the 
interval  CD  is  such  that  the  higher  orders  of  differences  may  be  neg- 
lected, the  values  of  the  forces  must  be  computed  for  the  successive 
dates  separated  by  the  interval  o>,  and  commencing  with  the  date 
tQ  —  \co  corresponding  to  the  argument  a  —  co,  tQ  being  the  date  to 
which  the  osculating  elements  belong.  Then,  since  the  last  terms 


..  .  .  .  , 

of  the  formulae  for  —^~,  —  j«p  and  —7^-  involve  ox,  oy,  and  02,  which 

are  the  quantities  sought,  the  subsequent  determination  of  the  differ- 
ential coefficients  must  be  performed  by  successive  trials.  Since  the 
integral  must  in  each  case  be  equal  to  zero  for  the  date  tQ,  it  will  be 
admissible  to  assume  first,  for  the  dates  t0  —  \co  and  tQ  -f-  \co  corre- 
sponding to  the  arguments  a  —  a>  and  a,  that  dx  —  0,  dy  =  0,  and 
dz  =  0,  and  hence  that  the  three  differential  coefficients,  for  each 


VARIATION   OF   CO-ORDINATEb.  445 

date,  are  respectively  equal  to  Xw  YQ,  and  Z0.  We  may  now  by  inte- 
gration derive  the  actual  or  the  very  approximate  values  of  the 
variations  of  the  co-ordinates  for  these  two  dates.  Thus,  in  the  case 
of  each  co-ordinate,  we  compute  the  value  of  ff(a  —  \co)  by  means 
of  the  equation  (26),  using  only  the  first  term,  and  the  value  of 
"f(a  —  to)  from  (36),  using  in  this  case  also  only  the  first  term.  The 
value  of  the  next  function  symbolized  by  "f  will  be  given  by 


Then  the  formula  (39),  putting  first  *  =  —  1  and  then  i  =  0,  and 
neglecting  second  differences,  will  give  the  values  of  the  variations 
of  the  co-ordinates  for  the  dates  a  —  co  and  a.  These  operations  will 
be  performed  in  the  case  of  each  of  the  three  co-ordinates;  and,  by 
means  of  the  results,  the  corrected  values  of  the  differential  coeffi- 
cients will  be  obtained  from  the  equations  (9),  the  value  of  3r  being 
computed  by  means  of  (7).  With  the  corrected  values  thus  derived 
a  new  table  of  integration  will  be  commenced  ;  and  the  values  of 
rf(a  —  \co)  and  "f(a  —  to)  will  also  be  recomputed.  Then  we  obtain, 
also,  by  adding  '/(a  —  £«;)  to  /(a),  the  value  of  ff(a  -f-  Jw),  and,  by 
adding  this  to  "f(a),  the  value  of  "f(a  -f  to). 

An  approximate  value  of  f(a  +  to)  may  now  be  readily  estimated, 
and  two  terms  of  the  equation  (39),  putting  i=  1,  will  give  an  ap- 
proximate value  of  the  integral.  This  having  been  obtained  for 
each  of  the  co-ordinates,  the  corresponding  complete  values  of  the 
differential  coefficients  may  be  computed,  and  these  having  been 
introduced  into  the  table  of  integration,  the  process  may,  in  a  similar 
manner,  be  carried  one  step  farther,  so  as  to  determine  first  approxi- 
mate values  of  3x,  8y,  and  dz  for  the  date  represented  by  the  argu- 
ment a  +  2«,  and  then  the  corresponding  values  of  the  differential 
coefficients.  We  may  thus  by  successive  partial  integrations  deter- 
mine the  values  of  the  unknown  quantities  near  enough  for  the  cal- 
culation of  the  series  of  differential  coefficients,  even  when  the  inte- 
grals are  involved  directly  in  the  values  of  the  differential  coefficients. 
If  it  be  found  that  the  assumed  value  of  the  function  is,  in  any  case, 
much  in  error,  a  repetition  of  the  calculation  may  become  necessary  ; 
but  when  a  few  values  have  been  found,  the  course  of  the  function 
will  indicate  at  once  an  approximation  sufficiently  close,  since  what- 
ever error  remains  affects  the  approximate  integral  by  only  one- 
twelfth  part  of  the  amount  of  this  error.  Further,  it  is  evident 
that,  in  cases  of  this  kind,  when  the  determination  of  the  values  of 
the  differential  coefficients  requires  a  preliminary  approximate  inte- 


446  THEORETICAL   ASTRONOMY. 

gratiou,  it  is  necessary,  in  order  to  avoid  the  effect  of  the  errors  in 
the  values  of  the  higher  orders  of  differences,  that  the  interval  (o 
should  be  smaller  than  when  the  successive  values  of  the  function  to 
be  integrated  are  already  known.  In  the  case  of  the  small  planets 
an  interval  of  40  days  will  afford  the  required  facility  in  the  approxi- 
mations; but  in  the  case  of  the  comets  it  may  often  be  necessary  to 
adopt  an  interval  of  only  a  few  days.  The  necessity  of  a  change  in 
the  adopted  value  of  co  will  be  indicated,  in  the  numerical  applica- 
tion of  the  formula?,  by  the  manner  in  which  the  successive  assump- 
tions in  regard  to  the  value  of  the  function  are  found  to  agree  with 
the  corrected  results. 

The  values  of  the  differential  coefficients,  and  hence  those  of  the 
integrals,  are  conveniently  expressed  by  adopting  for  unity  the  unit 
of  the  seventh  decimal  place  of  their  values  in  terms  of  the  unit  of 
space. 

165.  Whenever  it  is  considered  necessary  to  commence  to  take  into 
account  the  perturbations  due  to  the  second  and  higher  powers  of  the 
disturbing  force,  the  complete  equations  (14)  must  be  employed.  In 
this  case  the  forces  Jf,  Y9  and  Z  should  not  be  computed  at  once  for 
the  entire  period  during  which  the  perturbations  are  to  be  determined. 
The  values  computed  by  means  of  the  osculating  elements  will  be 
employed  only  so  long  as  simply  the  first  power  of  the  disturbing 
force  is  considered,  and  by  means  of  the  approximate  values  of  dx, 
dy,  and  dz  which  would  be  employed  in  computing,  for  the  next  place, 
the  last  terms  of  the  equations  (9),  we  must  compute  also  the  cor- 
rected values  of  JT,  F,  and  Z.  These  will  be  given  by  the  second 
members  of  (8),  using  the  values  of  x,  y,  and  z  obtained  from 


We  compute  also  q  from  (12),  and  then  from  Table  XVII.  find  the 
corresponding  value  of  /.  The  corrected  values  of  ,  2  ,  ,,2  >  and 
—Ej-  will  be  given  by  the  equations  (14),  and  these  being  introduced, 

in  the  continuation  of  the  table  of  integration,  we  obtain  new  values 
of  fix,  %,  and  dz  for  the  date  under  consideration.  If  these  differ 
much  from  those  previously  assumed,  a  repetition  of  the  calculation 
will  be  necessary  in  order  to  secure  extreme  accuracy.  In  this  repe- 
tition, however,  it  will  not  be  necessary  to  recompute  the  coefficients 
of  dx,  dy,  and  dz  in  the  formula  for  q,  their  values  being  given  with 
sufficient  accuracy  by  means  of  the  previous  assumption  ;  and  gene- 


VARIATION   OF   CO-ORDINATES.  447 

rally  a  repetition  of  the  calculation  of  X,  Y,  and  Z  will  not  be 
required. 

Next,  the  values  of  dx,  %,  and  dz  may  be  determined  approxi- 
mately, as  already  explained,  for  the  following  date,  and  by  mean* 
of  these  the  corresponding  values  of  the  forces  Jf,  F,  and  Z  will  be 
found,  and  also/  and  the  remaining  terms  of  (14),  after  which  the 
integration  will  be  completed  and  a  new  trial  made,  if  it  be  con- 
sidered necessary.  In  the  final  integration,  all  the  terms  of  the  for- 
mulae of  integration  which  sensibly  aifect  the  result  may  be  taken 
into  account.  By  thus  performing  the  complete  calculation  of  each 
successive  place  separately,  the  determination  of  the  perturbations  in 
the  values  of  the  co-ordinates  may  be  effected  in  reference  to  all 
powers  of  the  masses,  provided  that  we  regard  the  masses  and  co-or- 
dinates of  the  disturbing  bodies  as  being  accurately  known ;  and  it  is 
apparent  that  this  complete  solution  of  the  problem  requires  very 
little  more  labor  than  the  determination  of  the  perturbations  when 
only  the  first  power  of  the  disturbing  force  is  considered.  But 
although  the  places  of  the  disturbing  bodies  as  given  by  the  tables 
of  their  motion  may  be  regarded  as  accurately  known,  there  are  yet 
the  errors  of  the  adopted  osculating  elements  of  the  disturbed  body 
to  detract  from  the  absolute  accuracy  of  the  computed  perturbations; 
and  hence  the  probable  errors  of  these  elements  should  be  constantly 
kept  in  view,  to  the  end  that  no  useless  extension  of  the  calculation 
may  be  undertaken.  When  the  osculating  elements  have  been  cor- 
rected by  means  of  a  very  extended  series  of  observations,  it  will  be 
expedient  to  determine  the  perturbations  with  all  possible  rigor. 

When  there  are  several  disturbing  planets,  the  forces  for  all  of 

these  may  be  computed  simultaneously  and  united  in  a  single  sum, 

so  that  in  the  equations  (14)  we  shall  have  ZX,  SY,  and  2Z  instead 

of  X,  Y,  and  Z  respectively;  and  the  integration  of  the  expressions 

fJ^dx    d^ftii  d^ds 

for  —fif-,  —~j  and  —^-  will  then  give  the  perturbations  due  to  the 

Cit  Cut  ut 

action  of  all  the  disturbing  bodies  considered.  However,  when  the 
interval  to  for  the  different  disturbing  planets  may  be  taken  differently, 
it  may  be  considered  expedient  to  compute  the  perturbations  sepa- 
rately, and  especially  if  the  adopted  values  of  the  masses  of  some  of 
the  disturbing  bodies  are  regarded  as  uncertain,  and  it  is  desired  to 
separate  their  action  in  order  to  determine  the  probable  corrections 
to  be  applied  to  the  values  of  m,  mf,  &c.,  or  to  determine  the  effect 
of  any  subsequent  change  in  these  values  without  repeating  the  cal- 
culation of  the  perturbations. 


448 


THEORETICAL   ASTRONOMY. 


166.  EXAMPLE. — To  illustrate  the  numerical  application  of  the 
formulae  for  the  computation  of  the  perturbations  of  the  rectangular 
co-ordinates,  let  it  be  required  to  compute  the  perturbations  of 
Eurynome  @  arising  from  the  action  of  Jupiter  from  1864  Jan.  1.0 
Berlin  mean  time  to  1865  Jan.  15.0  Berlin  mean  time,  assuming  the 
osculating  elements  to  be  the  following : — 

Epoch  =  1864  Jan.  1.0  Berlin  mean  time. 
M9=      1°  29'    5".65 

7r0=   44    17  12  .17)  T 
r\         of\a    oo     K    £a  I  Ecliptic  and  Mean 

&60  =  ZUO      oa       O   .oy    * 

i,=     4    36  52.11. 
Po  =   11    15  51  .02 
log  a0=  0.3881319 


Equinox  1860.0 


From  these  elements  we  derive  the  following  values : — 


Berlin  Mean  Time.        x0          y0          z0 

Ios;r0 

1863  Dec. 

12.0 

+  1.53616 

+  1.23012 

—  0.03312 

0.294084, 

1864  Jan. 

21.0 

1.15097 

1.59918 

0.07369 

0.294837, 

March 

1.0 

0.69518 

1.87033 

0.10978 

0.300674, 

April 

10.0 

+  0.19817 

2.03141 

0.13936 

0.310864, 

May 

20.0 

—  0.31012 

2.08092 

0.16134 

0.324298, 

June 

29.0 

0.80326 

2.02602 

0.17523 

0.339745, 

Aug. 

8.0 

1.26055 

1.87959 

0.18122 

0.356101, 

Sept. 

17.0 

1.66729 

1.65711 

0.17990 

0.372469, 

Oct. 

27.0 

2.01414 

1.37473 

0.17209 

0.388214, 

Dec. 

6.0 

2.29597 

1.04766 

0.15870 

0.402894, 

1865  Jan. 

15.0 

—  2.51077 

+  0  68978 

—  0.14066 

0.416240. 

The  adopted  interval  is  a)  =  40  days,  and  the  co-ordinates  are  re- 
ferred to  the  ecliptic  and  mean  equinox  of  1860.0.  The  first  date, 
it  will  be  observed,  corresponds  to  tQ  —  \toy  and  the  integration  is  to 
commence  at  1864  Jan.  1.0. 

The  places  of  Jupiter  derived  from  the  tables  give  the  following 
values  of  the  co-ordinates  of  that  planet,  with  which  we  write  also 
the  distances  of  Eurynome  from  Jupiter  computed  by  means  of  the 
formula 

,.  =  (of  -  x?  +  <y  -  y)«  +  (z'  -  z)\ 


Berlin  Mean  Time.              a/ 

?' 

z' 

logr' 

logp 

1863 

Dec. 

12.0 

—  4 

.09683 

—  3. 

55184 

+  0.10533 

0.73425 

0 

.86866, 

1864 

Jan. 

21.0 

3 

.89630 

3. 

76053 

0.10152 

0.73368 

0 

.86713, 

March 

1.0 

3 

.68416 

3. 

95803 

0.09744 

0.73305 

0 

.86292, 

April 

10.0 

-3 

.46098 

4 

14366 

+  0.09304 

0.73237 

0 

.85622, 

NUMERICAL   EXAMPLE. 


449 


Berlin 

Mean 

Time. 

x' 

y' 

< 

logr' 

log/5 

1864 

May 

20.0 

—  3.22739 

—  4 

.31684 

+  0.08839 

0.73164 

0.84732, 

June 

29.0 

2. 

98405 

4 

.47693 

0.08346 

0.73086 

0.83656, 

Aug. 

8.0 

2. 

73162 

4 

.62343 

0.07827 

0.73003 

0.82428, 

Sept. 

17.0 

2. 

47085 

4.75576 

0.07284 

0.72915 

0.81077, 

Oct. 

27.0 

2. 

20247 

4 

.87345 

0.06720 

0.72823 

0.79628, 

Dec. 

6.0 

1. 

92728 

4.97606 

0.06134 

0.72726 

0.78098, 

1865 

Jan. 

15.0 

—  1. 

64600 

—  5 

.06301 

-f  0.05531 

0.72625 

0.76498. 

These  co-ordinates  are  also  referred  to  the  ecliptic  and  mean  equinox 
of  1860.0. 

If  we  neglect  the  mass  of  Eurynome  and  adopt  for  the  mass  of 
Jupiter 

,_         1 
~  1047.879' 

we  obtain,  in  units  of  the  seventh  decimal  place, 

«,«'  =  4518.27, 
and  the  equations  (9)  become 


cPfe 


=  4518.27 


Substituting  for  the  quantities  in  the  first  term  of  the  second  member 
of  each  of  these  equations  the  values  already  found,  we  obtain 


Argument. 

Date. 

6J2X0 

w2F0 

«% 

a  —  (o 

1863  Dec. 

12.0 

+  53.00 

+  47.09 

-  1.43, 

a 

1864  Jan. 

21.0 

53.71 

46.31 

0.91, 

a  -f-  to 

March 

1.0 

54.23 

45.18 

—  0.37, 

a  +  2u 

April 

10.0 

54.69 

43.59 

+  0.22, 

a  -f-  3&* 

May 

20.0 

55.23 

41.51 

0.70, 

it   i   ^\-(/j 

June 

29.0 

56.06 

38.96 

1.19, 

a  -f-  5o> 

Aug. 

8.0 

57.30 

35.92 

1.66, 

a  +  6«> 

Sept. 

17.0 

59.09 

32.47 

2.08, 

a  +  7" 

Oct. 

27.0 

61.55 

28.60 

2.43, 

a-{-8a> 

Dec. 

6.0 

64.85 

24.34 

2.69, 

a  +  9w 

1865  Jan. 

15.0 

+  69.09 

+  19.78 

-f  2.83, 

which  are  expressed  in  units  of  the  seventh  decimal  place. 

"We  now,  for  a  first  approximation,  regard  the  perturbations  as 


450  THEORETICAL   ASTRONOMY. 

being  equal  to  zero  for  the  dates  Dec.  12.0  and  Jan.  21.0,  and,  in  the 
case  of  the  variation  of  x,  we  compute  first 

'/(a  -  K>  =  -  A/'  (a  —  j«0  =  -  A  (53.71  -  53.00)  =  -  0.03, 


and  the  approximate  table  of  integration  becomes 

/(«  ~  «0  =  +  53.00  m_>}  __0  03  "/(«  -  *)  = 
/(«)          =  +  53.71  K  "/(a)          -= 

Then  the  formula  (39),  putting  first  i  =  —  1,  and  then  i  =  0,  gives 


Dec.  12.0  to=  +  2.24  +  -^p  =  +  6.66, 

Jan.  21.0  dx=  +  2.21  +  —-^  =  +  6.69. 

In  a  similar  manner,  we  find 

Dec.  12.0  Sy  =  +  5.85  dz  =  —  0.16, 

Jan.  21.0  fy  =  +  5.82  &  =  —  0.14. 

By  means  of  these  results  we  compute  the  complete  values  of  the 
second  members  of  equations  (40),  dr  being  found  from 


and  thus  we  obtain 

(PJoj  cPtJy  .cfrte 

Pate"  "^  "^  W^ 

Dec.  12.0        -|-  53.86        +  47.76        —  1.45        -f  8.85, 
Jan.  21.0        -f  54.23        +  47.25        —  0.96        +  8.63. 

We  now  commence  anew  the  table  of  integration,  namely, 

/         7        7          /        '/        7        /        '/      7 

+53.86  _  002+  2.26,  +47.76  ,    0  02  +  1.97,  -1.45  _0  Q2  -0.04, 

+54.23     5421+  2.24,  +47.25  ^  '      +  1.99,  -0.96  _Q98  -0.06, 

+56.45,  +49.26,  -1.04. 

the  formation  of  which  is  made  evident  by  what  precedes. 

We  may  next  assume  for  approximate  values  of  the  differential 
coefficients,  for  the  date  March  1.0,  +54.6,  +46.7,  and  —0.5, 
respectively;  and  these  give,  for  this  date, 


NUMERICAL   EXAMPLE.  451 

dx  =  -f  56.45  +  ~  =  +  61.00, 

1Z 

fy  =  +  49.26  +  ^.  =  +  53.15, 
fe  =  —   l.04~4^  =  —    1.08. 

By  means  of  these  approximate   values  we   obtain   the   following 
results  :  — 


1  864  March  1.0  "2=  +  55.01,    "2r=  +  53.86,    ^~  =  -  1.00, 

3r  =  +  71.03. 

Introducing  these  -into  the  table  of  integration,  we  find,  for  the  corre- 
sponding values  of  the  integrals, 

tx  =  +  61.03,  fy  =  +  53.75,  da  =  —  1.12. 

These  results  differ  so  little  from  those  already  derived  from  the 
assumed  values  of  the  function  that  a  repetition  of  the  calculation  is 
unnecessary.  This  repetition,  however,  gives 

55.04, 


Assuming,  again,  approximate  values  of  the  differential  coefficients 
for  April  10.0,  and  computing  the  corresponding  values  of  dx,  dy, 
and  dz,  we  derive,  for  this  date, 


=  +  48.06,  =  +  63.19, 


Introducing  these  into  the  table  of  integration,  and  thus  deriving 
approximate  values  of  dx,  dy,  and  dz  for  May  20,  we  carry  the  pro- 
cess one  step  further.  In  this  manner,  by  successive  approximations, 
we  obtain  the  following  results  :  — 


Date. 


1863  Dec.     12.0 

+  53.86 

+  47.76 

—  1.45, 

1864  Jan.      21.0 

54/23 

47.25 

0.96, 

March    1.0 

55.04 

53.91 

1.00, 

April    10.0 

48.06 

63.19 

1.54, 

May     20.0 

32.85 

65.40 

2.07, 

June    29.0 

16.74 

54.48 

1.75, 

Aug.       8.0 

8.62 

31.39 

—  0.36, 

Sept.     17.0 

+  14.20 

+    2.09 

+  1.86, 

452 


THEORETICAL   ASTRONOMY. 


Date.          *£ 

1864  Oct.  27.0  +  34.84 
Dec.  6.0     68.79 

1865  Jan.  15.0  +  112.64 


26.32 

47.87 
58.39 


w2^ 
+  4.44, 
6.86, 
+  8.68. 


The  complete  integration  may  now  be  effected,  and  we  may  use  both 
equation  (37)  and  equation  (39),  the  former  giving  the  integral  for 
the  dates  Jan.  1.0,  Feb.  10.0,  March  21.0,  &c.,  and  the  latter  the 
integrals  for  the  dates  in  the  foregoing  table  of  values  of  the  function. 
The  final  results  for  the  perturbations  of  the  rectangular  co-ordinates, 
expressed  in  units  of  the  seventh  decimal  place,  are  thus  found  to  be 
the  following : — 


Berlin  Mean  Time. 

6x 

6y 

9* 

1863  Dec.     12.0 

+  6.7 

+  5.9 

-0.2, 

1864  Jan.        1.0 

0.0 

0.0 

0.0, 

21.0 

+  6.8 

5.9 

0.1, 

Feb.     10.0 

27.1 

23.5 

0.5, 

March    1.0 

61.0 

53.7 

1.1, 

21.0 

108.9 

97.4 

2.0, 

April    10.0 

169.7 

155.7 

3.1, 

30.0 

242.7 

229.9 

4.7, 

May     20.0 

325.7 

320.3 

6.7, 

June       9.0 

417.1 

427.2 

9.3, 

29.0 

514.6 

549.1 

12.3, 

July     19.0 

616.1 

684.9 

15.7, 

Aug.      8.0 

720.8 

831.4 

19.5, 

28.0 

827.4 

986.0 

23.4, 

Sept.     17.0 

936.8 

1144.6 

27.0, 

Oct.        7.0 

1049.4 

1303.8 

30.2, 

27.0 

1168.2 

1460.0 

32.6, 

Nov.     16.0 

1295.4 

1609.4 

33.9, 

Dec.       6.0 

1435.6 

1749.6 

33.8, 

26.0 

1592.8 

1877.6 

32.0, 

1865  Jan.      15.0 

+  1^72.6 

+  1992.3 

—  28.2. 

During  the  interval  included  by  these  perturbations,  the  terms  of 
the  second  order  of  the  disturbing  forces  will  have  no  sensible  effect; 
but  to  illustrate  the  application  of  the  rigorous  formulae,  let  us  com- 
mence at  the  date  1864  Sept.  17.0  to  consider  the  perturbations  of 
the  second  order. 

In  the  first  place,  the  components  of  the  disturbing  force  must  be 
computed  by  means  of  the  equations 


NUMERICAL   EXAMPLE.  453 


*Z=  rfm'tf  (  Z-^  —  4,  V 
\     ^s          575  j 


The  approximate  values  of  Sx,  dy,  and  dz  for  Sept.  17.0  given  imme- 
diately by  the  table  of  integration  extended  to  this  date,  will  suffice 
to  furnish  the  required  values  of  the  disturbed  co-ordinates  by  means 
of 


and  to  find  p  =  pQ  +  dpy  we  have 


or 


in  which  ^0  is  the  modulus  of  the  system  of  logarithms.     Thus  we 
obtain,  for  Sept.  17.0, 

d  log  ^  =  +  0.0000084, 
<o*X=  +  59.09,  «>*  Y=  +  32.48,  «>*Z  =  +  2.08, 

which  require  no  further  correction. 
Next,  we  compute  the  values  of 


which  also  will  not  require  any  further  correction,  and  thus  we  form, 
according  to  (12),  the  equation 

q  =  —  0.29996&C  +  0.29815fy  —  0.03237&. 

The  approximate  values  of  dx,  dy,  and  dz  being  substituted  in  thia 
equation,  we  obtain 

q=  +  0.0000061, 

corresponding  to  which  Table  XVII.  gives 

log/=  0.477115. 
Hence  we  derive 

7.2  /M2P 

(fqx  -3x*)  =  -  44.87,         S3*  (fqy  -  9y)  =  -  30.40, 


454  THEORETICAL  ASTRONOMY. 

and  the  equations  (14)  give 

=  +  14.22,  =  +  2.08, 


These  values  being  introduced  into  the  table  of  integration,  the 
resulting  values  of  the  integrals  are  changed  so  little  that  a  repetition 
of  the  calculation  is  not  required. 

We  now  derive  approximate  values  of  dx,  %,  and  dz  for  Oct.  27.0, 
and  in  a  similar  manner  we  obtain  the  corrected  values  of  the  differ- 
ential coefficients  for  this  date ;  and  thus  by  computing  the  forces  for 
each  place  in  succession  from  approximate  values  of  the  perturbations, 
and  repeating  the  calculation  whenever  it  may  appear  necessary,  we 
may  determine  the  perturbations  rigorously  for  all  powers  of  the 
masses.  The  results  in  the  case  under  consideration  are  the  follow- 
ing:— 


J^/CttC/* 

dP 

dp 

dtz 

1864  Sept.  17.0 

+  14.22 

+  2.08 

+  1.87, 

Oct.  27.0 

34.84 

—  26.31 

4.44, 

Dec.  6.0 

68.77 

47.86 

6.86, 

1865  Jan.  15.0        +  112.60        —  58.39        +  8.68. 

Introducing  these  results  into  the  table  of  integration,  the  integrals 
for  Jan.  15.0  are  found  to  be 

fa  =  +  1772.6,        dy  =  -f  1992.3,        8z  =  —  28.2, 

agreeing  exactly  with  those  obtained  when  terms  of  the  order  of  the 
square  of  the  disturbing  forces  are  neglected. 

If  the  perturbations  of  the  rectangular  co-ordinates  referred  to  the 
equator  are  required,  we  have,  whatever  may  be  the  magnitude  of  the 

perturbations, 

dx,  =  dx, 

dyf  =  COS  e  dy  —  sin  e  dz,  (41 ) 

dzf  =  sin  e  dy  -f-  cos  e  dz, 

xn  y,,  z,  being  the  co-ordinates  in  reference  to  the  equator  as  the  fun 
damental  plane.     Thus  we  obtain,  for  1865  Jan.  15.0, 

dx,  =  +  1772.6,        dy,  =  +  1838.9,        8z,  ==  +  767.2. 

These  values,  expressed  in  seconds  of  arc  of  a  circle  whose  radius  is 
the  unit  of  space,  are 

dx=  +  36".562,        dy,  =  +  37".930,        dz,  =  +  15".825. 


VARIATION   OF   CO-ORDINATES.  455 

The  approximate  geocentric  place  of  the  planet  for  the  same  date  is 
a  =  183°  28',        S  =  —  5°  39',  log  A  =  0.3229, 

and  hence,  neglecting  terms  of  the  second  order,  we  derive,  by  means 
of  the  equations  (3)2,  for  the  perturbations  of  the  geocentric  right 
ascension  and  declination, 

Aa  =  —  17".03,  A*  =  +  5".67. 

167.  The  values  of  dx,  dy,  and  dz,  computed  by  means  of  the  co- 
ordinates referred  to  the  ecliptic  and  mean  equinox  of  the  date  t,  must 
be  added  to  the  co-ordinates  given  by  the  undisturbed  elements  and 
referred  to  the  same  mean  equinox.  The  co-ordinates  referred  to  the 
ecliptic  and  mean  equinox  of  t  may  be  readily  transformed  into  those 
referred  to  the  ecliptic  and  mean  equinox  of  another  date  t'.  Thus, 
let  #  denote  the  longitude  of  the  descending  node  of  the  ecliptic  of  tf 
on  that  of  £,  measured  from  the  mean  equinox  of  t,  and  let  TJ  be  the 
mutual  inclination  of  these  planes;  then,  if  we  denote  by  a?',  yf,  zr 
the  co-ordinates  referred  to  the  ecliptic  of  t  as  the  fundamental  plane, 
the  positive  axis  of  x9  however,  being  directed  to  the  point  whose 
longitude  is  6,  we  shall  have 

x'  =  x  cos  0  -f-  y  sin  0, 

(42) 


Let  us  now  denote  by  a?",  y",  z"  the  co-ordinates  when  the  ecliptic 
of  t  is  the  plane  of  xy,  the  axis  of  x  remaining  the  same  as  in  the 
system  of  x',  yf,  z'.  Then  we  shall  have 


y"  =  yf  cos  i)  —  z  sin  17,  (43) 

2"  =  y'  sin  f)  -f-  d  cos  ^. 

Finally,  transforming  these  so  that  the  axis  of  z  remains  unchanged 
while  the  positive  axis  of  x  is  directed  to  the  mean  equinox  of  t,  and 
denoting  the  new  co-ordinates  by  a?,,  y,,  zn  we  get 

.     x,  =  x"  cos  (0  +  p)  —  f  sin  (0  +  p\ 

y,  =  x"  sin  (0  +  p)  +  y"  cos  (0  +  p\  (44) 


«,=«", 


in  which  p  denotes  the  precession  during  the  interval  t'  —  t.  Elimi- 
nating x",  y",  and  z"  from  these  equations  by  means  of  (43)  and  (42), 
observing  that,  since  y  is  very  small,  we  may  put  cos  7  =  1,  we  get 


456  DHEOKETICAL    ASTRONOMY. 

x,  —  x  cosp  —  y  sin p  -f-  -  z  sin  (0  -f~  P)> 

8 

y,  —  xsmp-\-y  cosp  —  -  z  cos  (0  +  P)>  (45) 

s 

z,  =z  —  -  x  sin  6  -f  -  y  cos  0, 

9  S 

in  which  s  =  206264.8,  rj  being  supposed  to  be  expressed  in  seconds 
of  arc.  If  we  neglect  terms  of  the  order  j?3,  these  equations  become 

A^2  (V\  y* 

x,  =  x  —  ±^x  —  -  y  +  -  (sin  ^  +  P  cos  0)  2> 

"3  S  S 

&  =  y  — i|rY+f*- 7(008*— .p sin*)*,  (46) 

o  S  o 

2,  =  2  —  -  x  sin  0  +  -  y  cos  0. 

s  s 

These  formulae  give  the  co-ordinates  referred  to  the  ecliptic  and  mean 
equinox  of  one  epoch  when  those  referred  to  the  ecliptic  and  mean 
equinox  of  another  date  are  known.  For  the  values  of  p,  y,  and  0, 
we  have 

p  =  (50".21129  +  0".0002442966r)  (f  —  t\ 

TJ=(  (T.48892  —  0".000006143r)  (*'•-*), 

0  =  351°  36'  10"  +  39".79  (t  —  1750)  —  5".21  (if  —  0, 

in  which  r  =  \(tf  —  t)  —  1750,  t  and  t'  being  expressed  in  years  from 
the  beginning  of  the  era.  If  we  add  the  nutation  to  the  value  of  p, 
the  co-ordinates  will  be  derived  for  the  true  equinox  of  t'. 

The  equations  (45)  and  (46)  serve  also  to  convert  the  values  of  dxy 
%,  and  dz  belonging  to  the  co-ordinates  referred  to  the  ecliptic  and 
mean  equinox  of  t  into  those  to  be  applied  to  the  co-ordinates  re- 
ferred to  the  ecliptic  and  mean  equinox  of  t' .  For  this  purpose  it 
is  only  necessary  to  write  dx,  <%,  and  dz  in  place  of  x,  yy  and  z  re- 
spectively, and  similarly  for  x,,  yn  z,. 

In  the  computation  of  the  perturbations  of  a  heavenly  body  during 
a  period  of  several  years,  it  will  be  convenient  to  adopt  a  fixed  equi- 
nox and  ecliptic  throughout  the  calculation ;  but  when  the  perturba- 
tions are  to  be  applied  to  the  co-ordinates,  in  the  calculation  of  an 
ephemeris  of  the  body  taking  into  account  the  perturbations,  it  will 
be  convenient  to  compute  the  co-ordinates  directly  for  the  ecliptic 
and  mean  equinox  of  the  beginning  of  the  year  for  which  the 
ephemeris  is  required,  and  the  values  of  dx,  dy,  and  dz  must  be 
reduced,  by  means  of  the  equations  (45),  as  already  explained,  from 
the  ecliptic  and  mean  equinox  to  which  they  belong,  to  the  ecliptic 
and  mean  equinox  adopted  in  the  case  of  the  co-ordinates  required. 


VARIATION   OF   CO-ORDINATES.  457 

In  a  similar  manner  we  may  derive  formulae  for  the  transformation 
of  the  co-ordinates  or  of  their  variations  referred  to  the  mean  equinox 
and  equator  of  one  date  into  those  referred  to  the  mean  equinox 
and  equator  of  another  date;  but  a  transformation  of  this  kind  will 
rarely  be  required,  and,  whenever  required,  it  may  be  effected  by  first 
converting  the  co-ordinates  referred  to  the  equator  into  those  referred 
to  the  ecliptic,  reducing  these  to  the  equinox  of  t1  by  means  of  (45) 
or  (46),  and  finally  converting  them  into  the  values  referred  to  the 
equator  of  t'.  Since,  in  the  computation  of  an  ephemeris  for  the 
comparison  of  observations,  the  co-ordinates  are  generally  required 
in  reference  to  the  equator  as  the  fundamental  plane,  it  would  appear 
preferable  to  adopt  this  plane  as  the  plane  of  xy  in  the  computation 
of  the  perturbations,  and  in  some  cases  this  method  is  most  advan- 
tageous. But,  generally,  since  the  elements  of  the  orbit  of  the  dis- 
turbed planet  as  well  as  the  elements  of  the  orbits  of  the  disturbing 
bodies  are  referred  to  the  ecliptic,  the  calculation  of  the  perturbations 
will  be  most  conveniently  performed  by  adopting  the  ecliptic  as  the 
fundamental  plane.  The  consideration  of  the  change  of  the  position 
of  the  fundamental  plane  from  one  epoch  to  another  is  thus  also  ren- 
dered more  simple.  Whenever  an  ephemeris  giving  the  geocentric 
right  ascension  and  declination  is  required,  the  heliocentric  co-ordi- 
nates of  the  body  referred  to  the  mean  equinox  and  equator  of  the 
beginning  of  the  year  will  be  computed  by  means  of  the  osculating 
elements  corrected  for  precession  to  that  epoch,  and  the  perturbations 
of  the  co-ordinates  referred  to  the  ecliptic  and  mean  equinox  of  any 
other  date  will  be  first  corrected  according  to  the  equations  (46),  and 
then  converted  into  those  to  be  applied  to  the  co-ordinates  referred  to 
the  mean  equinox  and  equator.  If  the  perturbations  are  not  of  con- 
siderable magnitude  and  the  interval  t'  —  t  is  also  not  very  large,  the 
correction  of  dx,  dyy  and  dz  on  account  of  the  change  of  the  position 
of  the  ecliptic  and  of  the  equinox  will  be  insignificant;  and  the 
conversion  of  the  values  of  these  quantities  referred  to  the  ecliptic 
into  the  corresponding  values  for  the  equator,  is  effected  with  great 
facility. 

In  the  determination  of  the  perturbations  of  comets,  ephemerides 
being  required  only  during  the  time  of  describing  a  small  portion  of 
their  orbits,  it  will  sometimes  be  convenient  to  adopt  the  plane  of  the 
undisturbed  orbit  as  the  fundamental  plane.  In  this  case  the  posi- 
tive axis  of  x  should  be  directed  to  the  ascending  node  of  this  plane 
on  the  ecliptic,  and  the  subsequent  change  to  the  ecliptic  and  equinox, 
whenever  it  may  be  required,  will  be  readily  effected. 


458  THEORETICAL   ASTRONOMY. 

168.  The  perturbations  of  a  heavenly  body  may  thus  be  deter- 
mined rigorously  for  a  long  period  of  time,  provided  that  the  oscu- 
lating elements  may  be  regarded  as  accurately  known.  The  peculiar 
object,  however,  of  such  calculations  is  to  facilitate  the  correction  of 
the  assumed  elements  of  the  orbit  by  means  of  additional  observa- 
tions according  to  the  methods  which  have  already  been  explained; 
and  when  the  osculating  elements  have,  by  successive  corrections, 
been  determined  with  great  precision,  a  repetition  of  the  calculation 
of  the  perturbations  may  become  necessary,  since  changes  of  the  ele- 
ments which  do  not  sensibly  affect  the  residuals  for  the  given  differ- 
ential equations  in  the  determination  of  the  most  probable  corrections, 
may  have  a  much  greater  influence  on  the  accuracy  of  the  resulting 
values  of  the  perturbations. 

When  the  calculation  of  the  perturbations  is  carried  forward  for  a 
long  period,  using  constantly  the  same  osculating  elements,  —  and 
those  which  are  supposed  to  require  no  correction,  —  the  secular  per- 
turbations of  the  co-ordinates  arising  from  the  secular  variation  of 
the  elements,  and  the  perturbations  of  long  period,  will  constantly 
affect  the  magnitude  of  the  resulting  values,  so  that  fix,  Sy,  and  $2 
will  not  again  become  simultaneously  equal  to  zero.  Hence  it 
appears  that  even  when  the  adopted  elements  do  not  differ  much 
from  their  mean  values,  the  numerical  amount  of  the  perturbations 
may  be  very  greatly  increased  by  the  secular  perturbations  and  by 
the  large  perturbations  of  long  period.  But  when  the  perturbations 


are  large,  the  calculation  of  the  complete  values  of  ,  ,2  >  ,  2  >  and 
—j—-  (which  is  effected  indirectly)  cannot  be  performed  with  facility, 

requiring  often  several  repetitions  in  order  to  obtain  the  required 
accuracy,  since  any  error  in  the  value  of  the  second  differential  coeffi- 
cient produces,  by  the  double  integration,  an  error  increasing  propor- 
tionally to  the  time  in  the  values  of  the  integral.  Errors,  therefore, 
in  the  values  of  the  second  differential  coefficients  which  for  a  mode- 
rate period  would  have  no  sensible  effect,  may  in  the  course  of  a  long 
period  produce  large  errors  in  the  values  of  the  perturbations,  and  it 
is  evident  that,  both  for  convenience  in  the  numerical  calculation  and 
for  avoiding  the  accumulation  of  error,  it  will  be  necessary  from  time 
to  time  to  apply  the  perturbations  to  the  elements  in  order  that  the 
integrals  may,  in  the  case  of  each  of  the  co-ordinates,  be  again  equal 
to  zero.  The  calculation  will  then  be  continued  until  another  change 
of  the  elements  is  required. 


CHANGE   OF   THE   OSCULATING   ELEMENTS.  *59 

The  transformation  from  a  system  of  osculating  elements  for  one 
epoch  to  that  for  another  epoch  is  very  easily  effected  by  means  of 
the  values  of  the  perturbations  of  the  co-ordinates  in  connection 
with  the  corresponding  values  of  the  variations  of  the  velocities 

-T-,  -J-,  and  -rr-  The  latter  will  be  obtained  from  the  values  of  the 
at  at  at 

second  differential  coefficients  by  means  of  a  single  integration  ac- 
cording to  the  equations  (27)  and  (32).  Thus,  in  the  case  of  the 
example  given,  we  obtain  for  the  date  1865  Jan.  15.0,  by  means  of 
(32),  in  units  of  the  seventh  decimal  place, 

=  +  385.9,        40^  =  +  214.6,        40^  =  +  9.7. 
at  at 

The  velocities  in  the  case  of  the  disturbed  orbit  will  be  given  by  the 
formulae 

dx_  _dx0       d8x          dy dy0       d$y          dz  _  dz0       ddz      (     . 

"dt~~~3r~T"dT        ~dt~~~dt~~~dt'        ~dt '"" "dt '" W     ^    ' 

To  obtain  the  expressions  for  the  components  of  the  velocity 
resoHed  parallel  to  the  co-ordinates,  we  have,  according  to  the  equa- 
tions (6)2, 

dx        .        .    f  .         ,  dr  .  f  .         ,  dv 

-j-  =  sma  sm  ( A  -f  u)  —  -f-  r  sin  a  cos  ( A  -f-  u)  -j-, 
at  at  at 

-~  —  sin  b  sin  (5  -J-  u)  — r-  -f-  r  sin  b  cos  (L  -f-  u)  — =-» 
at  at  at 

dz        .        •    rsi  .     \dr   *       •  ff  ,     \  dv 

-jr  =  sm  c  sm  (  C  +  w)  -TT  +  r  sm  e  cos  (  C  -f-  u)  -j-- 

These  equations  are  applicable  in  the  case  of  any  Fundamental  plane, 
if  the  auxiliaries  sin  a,  sin  6,  sin  c,  A,  B,  and  C  are  determined  in 
reference  to  that  plane.  To  transform  them  still  further,  we  have 


rfr 
dt 


= 

dt 


in  which  to  denotes  the  angular  distance  of  the  J  »erihelion  from  the 
ascending  node.     Substituting  these  values,  we  ot  tain,  by  reduction, 


460  THEORETICAL   ASTRONOMY. 

dx 


—         /-       ((e  cos  <o  -f  cos  u)  cos  A  —  (e  sin  w  -j-  sin  w)  sin  A)  sin  a, 

(it  V    T) 

i- — ((e  cos  a>  -f-  cos  u)  cosB  —  (e  sin  w  -f-  sin  u)  sin  5)  sin  6, 


efe         kv\ -\-iMf,  N        /~»       /     •  •      <v   •    xv*   • 

-ir  = 7= ((e  cos  w  -L-  cos  «)  cos  C  —  (e  sm  at  -f  sm  w)  sin  C)  sm  c. 


Let  us  now  put 

"Hhwi 


(e  sin  a>  -f-  sin  w)  =  Fsin  U, 

(48) 

(e  cosa>  -f-  cos  if)  =  Fcos  i7, 
KJ» 
and  we  have 

—  =  Fsin  a  cos  (J.  -f  U"), 

-|-  =  Fsin  b  cos  (5  +  CO,  (49) 

dt 

^=Fsinccos(a4-  CO- 

These  equations  determine  the  components  of  the  velocity  of  a  hea- 
venly body  resolved  in  directions  parallel  to  the  co-ordinate  axes, 
and  for  any  fundamental  plane  to  which  the  auxiliaries  A,  B,  &c. 
belong.  When  the  ecliptic  is  the  fundamental  plane,  we  have 

sin  c  =  sin  i,  (7=0. 

The  sum  of  the  squares  of  the  equations  (48)  gives 


(*-«)' 


P 

and  hence  it  appears  that  F  is  the  linear  velocity  of  the  body. 

The  determination  of  the  osculating  elements  corresponding  to  any 
date  for  which  the  perturbations  of  the  co-ordinates  and  of  the  veloci- 
ties have  been  found,  is  therefore  effected  in  the  following  manner : — 

First,  by  means  of  the  osculating  elements  to  which  the  perturba- 
tions belong,  we  compute  accurate  values  of  r0,  a?0,  yQ,  z0,  and  by 

means  of  the  equations  (48)  and  (49)  we  compute  the  values  of  -~, 
-jp  and  -7£'  Then  we  apply  to  these  the  values  of  the  perturba- 
tions, and  thus  find  x,  y,  z,  -J-,  ~jt,  and  -^--  These  having  been 


CHANGE   OF   THE   OSCULATING   ELEMENTS.  461 


found,  the  equations  (32)!  will  furnish  the  values  of  ft,  i,  and  p; 
and  the  remaining  elements  may  be  determined  as  explained  in  Art* 
112.  Thus,  from 


Vr  sin  *0  =  kVp  (1 


dx    .      dy    ,      dz 


we  obtain  Vr  and     0  and  from 


r  sin  u  =  ( —  x  sin  ft  +  y  cos  ft)  sec  t, 
r  cos  it  =  a;  cos  ft 


we  derive  r  and  u;  and  hence  Ffrom  the  value  of  Vr.  When  i  is 
not  very  small,  we  may  use,  instead  of  the  preceding  expression  for 
r  sin  u, 

r  sin  u  =  z  cosec  i. 

Next,  we  compute  a  from 

2a  —  r  =  ; 


and  from 

2ae  sin  o>  =  —  (2a  —  r)  sin  (2^0  -}-  w)  —  r  sin  w, 
2ae  cos  a>  =  —  (2a  —  r)  cos  (2^0  +  M)  —  r  cos  w> 

we  find  to  and  e.  The  mean  daily  motion  and  the  mean  anomaly  or 
the  mean  longitude  for  the  epoch  will  then  be  determined  by  means 
of  the  usual  formulae. 

In  the  case  of  a  very  eccentric  orbit,  after  r  and  u  have  been  found, 

-r-  will  be  given  by  equations  (48)6,  and  the  values  of  e  and  v  will 

be  given  by  the  equations  (49)6.  Then  the  perihelion  distance  will 
be  found  from 

P 


and  the  time  of  perihelion  passage  will  be  found  from  v  and  e  by 
means  of  Table  IX.  or  Table  X. 

In  the  numerical  values  of  the  velocities  -rr»  -77,  &c.,  more  decimals 

at     at 

must  be  retained  than  in  the  values  of  the  co-ordinates,  and  enough 
must  be  retained  to  secure  the  required  accuracy  of  the  solution.  If 
it  be  considered  necessary,  the  different  parts  of  the  calculation  may 
be  checked  by  means  of  various  formulse  which  have  already  been 
given.  Thus,  the  values  of  ft  and  i  must  satisfy  the  equation 


462  THEORETICAL   ASTRONOMY. 

z  cos  i  —  y  sin  i  cos  &  -f  x  sin  i  sin  &  =  0. 
We  have,  also, 


r'^  +  ^-f  z\ 
z  =  r  sin  it  sin  i, 

which  must  be  satisfied  by  the  resulting  values  of  F,  r,  and  u;  and 
the  values  of  a  and  e  must  satisfy  the  equation 

p  =  a  (1  —  e2)  —  a  cos*  ?>. 

169.  When  the  plane  of  the  undisturbed  orbit  is  adopted  as  the 
fundamental  plane,  we  obtain  at  once  the  perturbations 

d  (r  cos  u),  8  (r  sin  u),  fa, 

and  from  these  the  perturbations  of  the  polar  co-ordinates  are  easily 
derived.  There  are,  however,  advantages  which  may  be  secured  by 
employing  formulae  which  give  the  perturbations  of  the  polar  co-or- 
dinates directly,  retaining  the  plane  of  the  orbit  for  the  date  t0  as  the 
fundamental  plane. 

Let  w  denote  the  angle  which  the  projection  of  the  disturbed 
radius-vector  on  the  plane  of  xy  makes  with  the  axis  of  x,  and  /9  the 
latitude  of  the  body  with  respect  to  the  plane  of  xy;  then  we  shall 

have 

x  =  r  cos  ft  cos  w, 

y  =  r  cos  ft  sin  w,  (50) 

z  =  r  sin  ft. 

Let  us  now  denote  by  X,  Yy  and  Zj  respectively,  the  forces  which  are 
expressed  by  the  second  members  of  the  equations  (1),  and  the  first 
two  of  these  equations  give 


C  being  the  constant  of  integration.     The  equations  (50)  give 


dx  d(rcosft}  Q  .       dw 

-—  =  cos  w         j.  —  -  —  r  cos  ft  sm  w  -r 
dt  dt  at 


dy        .       d(rcoaft)    .  Q          dw 

~-  =  smw         j.  —  -  -f  r  cos  ft  cosw  -=-, 
dt  at  at 

and  hence 

dy          dx  .    dw 

-  - 


VARIATION   OF   POLAR   CO-ORDINATES.  463 

Therefore  we  have 

r'cos2£^T=  £(Yx  —  Xy)dt+  C. 
ut       %/ 

If  we  denote  by  S0  the  component  of  the  disturbing  force  in  a  direc- 
tion perpendicular  to  the  disturbed  radius-vector  and  parallel  with 
the  plane  of  xy,  we  shall  have 

X  =  —  S0  sin  iv,  Y=  S0  cos  w, 

and 

Therefore 

r2cos2/5^-=  Cs9r<x*pdt+  C. 

at       J 

In  the  undisturbed  orbit  we  have  /9  =  0,  and 


and  thus  the  preceding  equation  becomes 

r2  cos2  ft       =     $  r  cos  ft  dt  +  IcVp.  (1  +  ro).  (51) 


The  equations  (1)  also  give 
-f-  y^2y  +  zfflz 


==  ^     •    y     •  ^ 
r  dt*  r2  r          r         r 

If  we  denote  by  R  the  component  of  the  disturbing  force  in  thp 
direction  of  the  disturbed  radius-vector,  we  have 


We  have,  also, 


R  =  X-  -1-  Yy-  +  Z*-.  (53) 

r  r          r 


xd*x  +  yd*y  -f  zd*z  —  d  (xdx  -f  ydy  +  2^2)  —  (<fo2  +  c?2/J  -f- 
=  d  (rdr)  —  (dr*  -f  r2Jv2)  —  rdV  —  rW, 


f?   denoting  the   true    anomaly   in    the   disturbed    orbit,   or,   since 


-f  yd*y  +  2C?22  =  r<Fr  —  r1  cos2  /5  duP  — 
Hence  the  equation  (52)  becomes 


dw*         dp  ,   P(l-f  m)       0  .... 

--  — rp-f     v  <^=J?.  154) 


464  THEORETICAL   ASTRONOMY. 

170.  The  equations  (51)  and  (54),  in  connection  with  the  last  of 
equations  (1),  completely  represent  the  motion  of  a  heavenly  body 
about  the  sun  when  acted  upon  by  disturbing  forces,  and,  when  com- 
pletely integrated,  they  will  give  the  values  of  w,  r,  and  z  for  any 
point  of  the  orbit;  but,  since  they  cannot  be  integrated  directly,  we 
must,  as  in  the  case  of  the  rectangular  co-ordinates,  find  the  equations 
which  give  by  integration  the  values  of  dw,  3r,  and  z.  In  the  case 
of  the  undisturbed  orbit,  we  have 


d\          dwf        fr(l+m) 
df         °-d?   '  r02 

If  we  denote  by  dw  the  variation  of  w  arising  from  the  action  of  the 
disturbing  force,  we  have  w  =  WQ  +  dw;  and  hence  we  easily  find, 
from  (51), 


r2  cos2 
We  have,  further, 
which  gives 

Let  us  now  put 


.  (56) 


r'cos'/?' 
and  we  have 

9 

(58) 


The  equation  (56),  therefore,  becomes 
in  which  we  put 


,fio 


If  we  substitute  r0  +  dr  for  r  in  equation  (54),  and  combine  the 
result  with  the  second  of  equations  (55),  we  get 


VARIATION   OF   POLAR   CO-ORDINATES.  465 

and  if  we  put 

9"  =  njhi^r>  f'g"=l-^,  (61) 

we  have 

'"=rw-  (63) 

and  hence 


Finally,  we  have,  from  the  last  of  equations  (1), 

db  *'(l+m) 

3?  =  *-      -?  --  *>  (64) 

by  means  of  which  the  value  of  z  may  be  found,  since,  in  the  case  of 
the  undisturbed  motion,  we  have  z0  =  0. 

The  values  of/'  corresponding  to  different  values  of  q'  may  be 
tabulated  with  the  argument  <?',  and,  since  the  equation  (62)  is  of  the 
same  form  as  (58),  the  same  table  will  give  the  value  of/"  when  q" 
is  used  as  the  argument.  Table  XVII.  gives  the  values  of  log/'  or 
log/"  corresponding  to  values  of  qf  or  q"  from  —  0.03  to  -f-  0.03. 
Beyond  the  limits  of  this  table  the  required  quantities  may  be  com- 
puted directly. 

171.  When  we  consider  only  terms  of  the  first  order  with  respect 
to  the  disturbing  force,  we  have 


and  the  equations  become 

~j —  —  "I  I  Bffodt —  &Tt 

at         r0  J  rQ 

O~        x.  /  O7.2/r1        I       ^^  V 

(65) 


d*z        „      f  (1  +  m) 
~d^~~  ~^~ 

In  determining  the  perturbations  of  a  heavenly  body,  we  first  con- 
sider only  the  terms  depending  on  the  first  power  of  the  disturbing 
force,  for  which  these  equations  will  be  applied.  The  value  of  dr 

30 


466  THEORETICAL   ASTRONOMY. 

will  be  obtained  from  the  second  equation  by  an  indirect  process,  as 
already  illustrated  for  the  case  of  the  variation  of  the  rectangular 
co-ordinates.  Then  dw  will  be  obtained  directly  from  the  first 
equation,  and,  finally,  z  indirectly  from  the  last  equation.  Each  of 
the  integrals  is  equal  to  zero  for  the  date  tw  to  which  the  osculating 
elements  belong. 

When  the  magnitude  of  the  perturbations  is  such  that  the  terms 
depending  on  the  squares  and  products  of  the  masses  must  be  con- 
sidered, the  general  equations  (59),  (63),  and  (64)  will  be  applied. 
The  values  of  the  perturbations  for  the  dates  preceding  that  for 
which  the  complete  expressions  are  to  be  used,  will  at  once  indicate 
approximate  values  of  dw,  dr,  and  z;  and  with  the  values 

r  =  rQ  -j-  drt  w  =  w9-{-  dw,  sin  /5  =  _, 

the  components  of  the  disturbing  force  will  be  computed.  We  compute 
also  qf  from  the  first  of  equations  (57),  and  q"  from  the  first  of  (61); 
then,  by  means  of  Table  XVII.,  we  derive  the  corresponding  values 
of  log/'  and  log/".  The  coefficients  of  dr  in  the  expressions  for 
q  and  q"  will  be  given  with  sufficient  accuracy  by  means  of  the 
approximate  values  of  dr  and  sin  /?,  and  will  not  require  any  further 
correction.  Then  we  compute  8Qr  cos/9,  and  find  the  integral 


CjS 


and  the  complete  value  of  -^~  will  be  given  by  (59).  The  value 
of  —Tp-  will  then  be  given  by  equation  (63).  The  term  r  I  -57  )  will 

always  be  small,  and,  unless  the  inclination  of  the  orbit  of  the  dis- 
turbed body  is  large,  it  may  generally  be  neglected.  Whenever  it  shall 

be  required,  we  may  put  it  equal  to  -  (  "JT  /  •    The  corrected  values 

of  the  differential  coefficients  being  introduced  into  the  table  of  inte- 
gration, the  exact  or  very  approximate  values  of  3w,  dr,  and  z  will 
be  obtained.  Should  these  results,  however,  differ  much  from  the 
corresponding  values  already  assumed,  a  repetition  of  the  calculation 
may  become  necessary.  In  this  manner,  by  computing  each  place 
separately,  the  terms  depending  on  the  squares,  products,  and  higher 
powers  of  the  disturbing  forces  may  be  included  in  the  results.  It 
will,  however,  be  generally  possible  to  estimate  the  values  of  dw,  dr, 


VARIATION   OF   POLAR   CO-ORDINATES.  467 

and  2  for  two  or  three  intervals  in  advance  to  a  degree  of  approxi- 
mation sufficient  for  the  computation  of  the  forces  for  these  dates. 

In  order  that  the  quantity  co,  representing  the  interval  adopted  in 
the  calculation  of  the  perturbations,  may  not  appear  in  the  integra- 
tion, we  should  introduce  it  into  the  equations  as  in  the  case  of  the 
variation  of  the  rectangular  co-ordinates.  Thus,  in  the  determina- 
tion of  3w  we  compute  the  values  of  ^—rr^  and  since  the  second 
member  of  the  equation  contains  the  integral  \  S0r  cos  /9  ctt,  if  we 

introduce  the  factor  a>2  under  the  sign  of  integration,  this  integral, 
omitting  the  factor  at  in  the  formulae  of  integration,  will  become 

(*)J  S0r  cos  /9  dtj  as  required.     The  last  term  of  the  equation  will  be 

multiplied  by  a). 

In  the  case  of  dr,  each  term  of  the  equation  for  — — -  must  contain 

(Jut 

the  factor  co2.  If  the  second  of  equations  (65)  is  employed,  the  first 
and  third  terms  of  the  second  member  will  be  multiplied  by  co2-,  but 
since  the  value  of  SQ  is  supposed  to  be  already  multiplied  by  o>2,  the 
second  term  will  only  be  multiplied  by  a). 

The  perturbations  may  be  conveniently  determined  either  in  units 
of  the  seventh  decimal  place,  or  expressed  in  seconds  of  arc  of  a 
circle  whose  radius  is  unity.  If  they  are  to  be  expressed  in  seconds, 
the  factor  s  =  206264.8  must  be  introduced  so  as  to  preserve  the 
homogeneity  of  the  several  terms,  and  finally  dr  and  dz  must  be  con- 
verted into  their  values  in  terms  of  the  unit  of  space. 

172.  It  remains  yet  to  derive  convenient  formula  for  the  deter- 
mination of  the  forces  S0,  R,  and  Z.  For  this  purpose,  it  first  becomes 
necessary  to  determine  the  position  of  the  orbit  of  the  disturbing 
planet  in  reference  to  the  fundamental  plane  adopted,  namely,  the 
plane  defined  by  the  osculating  elements  of  the  disturbed  orbit  at  the 
instant  1Q.  Let  V  and  & '  denote  the  inclination  and  the  longitude  of 
the  ascending  node  of  the  disturbing  body  with  respect  to  the  ecliptic, 
and  let  /denote  the  inclination  of  the  orbit  of  the  disturbing  body 
with  respect  to  the  fundamental  plane.  Further,  let  N  denote  the 
longitude  of  its  ascending  node  on  the  same  plane  measured  from  the 
ascending  node  of  this  plane  on  the  ecliptic  or  from  the  point  whose 
longitude  is  &0,  and  let  Nf  be  the  angular  distance  between  the  as- 
cending node  of  the  orbit  of  the  disturbing  body  on  the  ecliptic  and 
the  ascending  node  on  the  fundamental  plane  adopted.  Then,  from 
the  spherical  triangle  formed  by  the  intersection  of  the  plane  of  the 


THEORETICAL   ASTRONOMY. 


ecliptic,  the  fundamental  plane,  and  the  plane  of  the  orbit  of  the  dis- 
turbing body  with  the  celestial  vault,  we  have 

sin  ^Isin  J  (.y  +  N')  =  sin  £  (&'  -  £0)  sin  J  (i'  +  i0), 


from  which  to  find  N9  Nf,  and  /. 

Let  ft'  denote  the  heliocentric  latitude  of  the  disturbing  planet 
with  respect  to  the  fundamental  plane,  wr  its  longitude  in  this  plane 
measured  from  the  axis  of  x,  as  in  the  case  of  w,  and  uQf  the  argu- 
ment of  the  latitude  with  respect  to  this  plane.  Then,  according  to 
the  equations  (82)w  we  have 

tan  (V  —  N)  =  tan  u0'  cos  I, 


If  ur  denotes  the  argument  of  the  latitude  of  the  disturbing  planet 
with  respect  to  the  ecliptic,  we  have 

u0'  =  u'  —  N'.  (68) 

This  formula  will  give  the  value  of  u0f,  and  then  wf  and  ft'  will  be 
found  from  (67).     We  have,  also, 

cos  uQr  =  cos  ft  cos  (w'  —  N\ 

which  will  serve  to  indicate  the  quadrant  in  which  wr  —  J\Tmust  be 
taken. 

The  relations  here  derived  are  evidently  applicable  to  the  case  in 
which  the  elements  of  the  orbits  of  the  disturbed  and  disturbing 
planets  are  referred  to  the  equator,  the  signification  of  the  quantities 
involved  being  properly  considered. 

«  The  co-ordinates  of  the  disturbing  planet  in  reference  to  the  plane 
of  the  disturbed  orbit  at  the  instant  t0  as  the  fundamental  plane  will 
be  given  by 

x'  =  r'  cos  ft  cos  w'j 

i/  =  r'  cospsinw',  (691 


To  find  the  force  R,  we  have 

R  =  X-+  Yy-  +  Z-. 
r  '       r  '      r 


VARIATION   OF   POLAK  CO-ORDINATES.  469 

and 


Substituting  in  these  the  values  of  x't  yf,  z'  given  by  (69),  and  the 
corresponding  values  of  x,  yy  z  given  by  (50),  and  putting 

*  =  ^~£'  (70) 

we  get 
.      R  =  m'k*  I  h  r'  cos  /3'  cos  /?  cos  (w  —  w)  +  /i  r'  sin  £  sin  /5'  —  -^  ).  (71) 

The  equation 

S0rcosj3=  Yx-  Xy 
gives 

SQ  =  m'k*  h  r'  cos  f  sin  (wr  —  w),  (72) 

from  which  to  find  S0.     Finally,  we  have 

Z=mfk*  lhrf  sin  p  —  -8  ),  (73) 

from  which  to  find  Z. 

When  we  determine  the  perturbations  only  with  respect  to  the 
first  power  of  the  disturbing  force,  the  expressions  for  R,  SQ,  and  Z 
become 

jB  =  w'*i(  hr'  cospcosW  —  wJ—^}, 

\  Po  I  (74) 

S0  =  m'k*  h  r'  cos  jf  sin  (wf  —  wj, 
Z  = 


To  compute  the  distance  p,  we  have 

,«  =  (-,/  _  „.)«  +  (y'  - 
which  gives 


^j  _  r'»  _j_  r2  _  2r  /  cos  ^9  cos  ft  cos  («/  —  w)  —  2r  r'  sin  £  sin  ^,  (75) 
and,  if  we  neglect  terms  of  the  second  order,  we  have 

P*  =  r'2  +  r02  —  2r0  /  cos  p  cos  (wf  —  w0).  (76) 

If  we  put 

cos  r  =  cos  £  cos  £'  cos  (w'  —  w)  +  sin  0  sin  f,  (77) 

we  have 

—  2rr'cosr 

+  (r  —  r'  cos  r)8; 


470  THEOEETICAL   ASTKONOMY. 

and  hence  we  may  readily  find  p  from 

^     ' 


the  exact  value  of  the  angle  n,  however,  not  being  required. 
Introducing  f  into  the  expression  for  R,  it  becomes 


(79) 


by  means  of  which  R  may  be  conveniently  determined. 


173.  When  we  neglect  the  terms  depending  on  the  squares  and 
higher  powers  of  the  masses  in  the  computation  of  the  perturbations, 
the  forces  jR,  SQ,  and  Z  will  be  computed  by  means  of  the  equations 
(74),  pQ  being  found  from  (76)  or  from  (78),  when  we  put 

cos  f  =  cos  p  cos  (wf  —  w0). 

But  when  the  terms  of  the  order  of  the  square  of  the  disturbing 
force  are  to  be  taken  into  account,  the  complete  equations  must  be 
used.  Thus,  we  find  p  from  (78),  S0  from  (72),  Z  from  (73),  and  R 
from  (71)  or  (79).  The  values  of  dw,  dr,  and  z,  computed  to  the 
point  at  which  it  becomes  necessary  to  consider  the  terms  of  the 
second  order,  will  enable  us  at  once  to  estimate  the  values  of  the 
perturbations  for  two  or  three  intervals  in  advance  to  a  degree  of 
approximation  sufficient  for  the  calculation  of  the  forces;  and  the 
values  of  R,  8Q,  and  Z  thus  found  will  not  require  any  further  cor- 
rection. 

When  the  places  of  the  disturbing  planet  are  to  be  derived  from 
an  ephemeris  giving  the  heliocentric  longitudes  and  latitudes,  the 
values  of  &  '  and  ir  will  be  obtained  from  two  places  separated  by  a 
considerable  interval,  and  then  the  values  of  uf  will  be  determined 
by  means  of  the  first  of  equations  (82)L  or  by  means  of  (85)x.  When 
the  inclination  V  is  very  small,  it  will  be  sufficient  to  take 

u'  =  l'—R'  +  8  tan2  Jt*  sin  2  (lr  -  ft'), 

in  which  s  =  206264.8.  But  when  the  tables  give  directly  the  lon- 
gitude in  the  orbit,  uf  +  &',  by  subtracting  &'  from  each  of  these 
longitudes  we  obtain  the  required  values  of  ur. 

It  should  be  observed,  also,  that  the  exact  determination  of  the 
values  of  the  forces  requires  that  the  actual  disturbed  values  of  r', 
10',  and  /?'  should  be  used.  The  disturbed  radius-vector  r'  will  be 


VARIATION   OF   POLAR   CO-ORDINATES.  471 

given  immediately  by  the  tables  of  the  motion  of  the  disturbing 
body,  but  the  determination  of  the  actual  values  of  w'  and  ft'  re- 
quires that  we  should  use  the  actual  values  of  N',  Ny  and  I  in  the 
solution  of  the  equations  (68)  and  (67).  Hence  the  disturbed  values 
of  &  '  and  if  should  be  used  in  the  determination  of  these  quantities 
for  each  date  by  means  of  (66).  It  will,  however,  generally  be  tho 
case  that  for  a  moderate  period  the  variation  of  &'  and  if  may  be 
neglected;  and  whenever  the  variation  of  either  of  these  has  a  sensi- 
ble effect,  we  may  compute  new  values  of  N9  Nr,  and  /  from  time  to 
time,  by  means  of  which  the  true  values  may  be  readily  interpolated 
for  each  date.  We  may  also  determine  the  variations  of  N,  Nf,  and 
/  arising  from  the  variation  of  &'  and  i1,  by  means  of  differential 
formulae.  Thus  the  relations  will  be  similar  to  those  given  by  the 
equations  (71)2,  so  that  we  have 

„  sin-ZV' 


.     ,_.,  -  ; 

sin(£'—  Q.)  smJ 


31   =  sin  N'  sin  i'  $&'  -f  cos  N'  8ir, 

fh>m  which  to  find  dN',  3N,  and  81. 

When  the  perturbations  are  computed  only  in  reference  to  the  first 
power  of  the  mass,  the  change  of  Qf  and  if  may  be  entirely  neg- 
lected; but  when  the  perturbations  are  to  be  computed  for  a  long 
period  of  time,  and  the  terms  depending  on  the  squares  and  products 
of  the  disturbing  forces  are  to  be  included,  it  will  be  advisable  to 
take  into  account  the  values  of  dN,  3Nf,  and  dl,  and,  using  also  the 
value  of  u'  in  the  actual  orbit  of  the  disturbing  body,  compute  the 
actual  values  of  w'  and  ft'. 

In  the  case  of  several  disturbing  bodies,  the  forces  will  be  deter- 
mined for  each  of  these,  and  then,  instead  of  R,  S0,  and  Z,  in  the 
formulae  for  the  differential  coefficients,  2R,  28W  and  2  Z  will  be  used. 

174.  By  means  of  the  values  of  dw,  fir,  and  z,  the  heliocentric  or 
(he  geocentric  place  of  the  disturbed  planet  may  be  readily  found. 
Thus,  let  the  positive  axis  of  x  be  directed  to  the  ascending  node  of 
the  osculating  orbit  at  the  instant  t0  on  the  plane  of  the  ecliptic; 
then,  in  the  undisturbed  orbit,  we  shall  have 


u  denoting  the  argument  of  the  latitude.     Let  #„  y,,  z,  be  the  co-or- 


472  THEORETICAL   ASTRONOMY. 

dinates  of  the  body  referred  to  a  system  of  rectangular  co-ordinates 
in  which  the  ecliptic  is  the  plane  of  xy,  and  in  which  the  positive 
axis  of  x  is  directed  to  the  vernal  equinox.  Then  we  shall  have 

x,  =  x  cos  &0  —  y  cosi0 sin  &0  +  z  sin iQ  sin  £0, 
y,  =  x  sin  &0  +  y  cosiQ  cos  &0  —  *  sin  iQ  cos  &0, 
z,=y  sin  i0  +  z  cos  t0, 

or,  introducing  the  values  of  x  and  y  given  by  (50), 

Xf=r  cos  /9  cos  w  cos  &0  —  r  cos  ft  sin  w  cos i0  sin  £^0  -f-  2  sin  i0  sin  £^0, 

y,  =r  cos  /?  cos  w  sin  Q0  -f-  r  cos  /5  sin  w  cos  i0  cos  &0  —  z  sin  i0  cos  &0,  (81) 

zt  =r  cos  /?  sin  w  sin  i0  -f-  2  cos  i0. 

Introducing  also  the  auxiliary  constants  for  the  ecliptic  according  to 
the  equations  (94^  and  (96)w  we  obtain 

xt  =  r  cos  /?  sin  a  sin  ( J.  -(-  w)  -f-  2  cos  a, 

y,  =r  cos  /3  sin  6  sin  (^  -f  w)  -f  z  cos  6,  (82) 

2,  =  r  cos  /?  sin  i0  sin  w  +  2  cos  i0, 

by  means  of  which  the  heliocentric  co-ordinates  in  reference  to  the 
ocliptic  may  be  determined. 

If  the  place  of  the  disturbed  body  is  required  in  reference  to  the 
equator,  denoting  the  heliocentric  co-ordinates  by  xfn  y,,y  z,,,  and  the 
obliquity  of  the  ecliptic  by  e,  we  have 

xn  =  x, 

y,,=yfcose  —  z,  sine, 

zn  =  yf  sin  e  -f-  z,  cos  e. 

Substituting  for  xn  yn  z,  their  values  given  by  (81),  and  introducing 
the  auxiliary  constants  for  the  equator,  according  to  the  equations 
'99)x  and  (101)w  we  get 

xn  =  r  cos  /?  sin  a  sin  (A  -f-  w}  -\-  z  cos  a, 

ylt  =  r  cos  /3  sin  b  sin  (B  -f-  w)  -f-  z  cos  b,  (83) 

zt,  =  r  cos  ft  sin  c  sin  ( (7  -j-  w)  -\-  z  cos  c. 

The  combination  of  the  values  derived  from  these  equations  with  the 
corresponding  values  of  the  co-ordinates  of  the  sun,  will  give  the 
required  geocentric  places  of  the  disturbed  body.  These  equations 
are  applicable  to  the  case  of  any  fundamental  plane,  provided  that 
the  auxiliary  constants  a,  A,  b,  jB,  &c.  are  determined  with  respect 
to  that  plane.  In  the  numerical  application  of  the  formulae,  the 
value  of  w  will  be  found  from 

w  =  it,  -f-  dwt 


VAEIATION   OF   POLAR   CO-ORDINATES.  473 

UQ  being  the  argument  of  the  latitude  for  the  fundamental  osculating 
elements,  and  care  must  be  taken  that  the  proper  algebraic  sign  is 
assigned  to  cos  a,  cos  b,  and  cos  c. 

If  the  values  of  TTO,  &0,  and  iQ  used  in  the  calculation  of  the  per- 
turbations are  referred  to  the  ecliptic  and  mean  equinox  of  the  date 
t0',  and  the  rectangular  co-ordinates  of  the  disturbed  body  are  required 
in  reference  to  the  ecliptic  and  mean  equinox  of  the  date  t0",  the 
value  of  w  must  be  found  from 


the  value  of  WQ  referred  to  the  ecliptic  of  t0f  being  reduced  to  that  of 
£0",  by  means  of  the  first  of  equations  (115)^  Then  &0  and  iQ  should 
be  reduced  from  the  ecliptic  and  mean  equinox  of  t0'  to  the  ecliptic 
and  mean  equinox  of  tQ"  by  means  of  the  second  and  third  of  the 
equations  (115)^  and,  using  the  values  thus  found  in  the  calculation 
of  the  auxiliary  constants  for  the  ecliptic,  the  equations  (82)  will 
give  the  required  values  of  the  heliocentric  co-ordinates.  If  the  co- 
ordinates referred  to  the  mean  equinox  and  equator  of  the  date  tQ" 
are  to  be  determined,  the  proper  corrections  having  been  applied  to 
QQ  and  iw  the  mean  obliquity  of  the  ecliptic  for  this  date  will  be 
employed  in  the  determination  of  the  auxiliary  constants  a,  J.,  &c. 
with  respect  to  the  equator,  and  the  equations  (83)  will  then  give 
the  required  values  of  the  co-ordinates. 

If  we  differentiate  the  equations  (83),  we  obtain,  by  reduction, 

-^  =  r  cos  ft  sin  a  cos  ( A  -f  w)  —rr  -f-  sec  ft  sin  a  sin  ( A  -j-  w)  —j- 

-f  (cos  a  —  tan  ft  sin  a  sin  (A  +  w))  -rrt 

^-  =  r  cos  ft  sin  b  cos  (B  -j-  w)  -7-  -j-  sec  ft  gin  &  sm  C^  +  tu)  -jr 

dz    (84) 

-f  (cos  b  —  tan  ft  sin  b  sin  (J5  +w))  -77. 

A'  =rcos/5sinccos((7-f  w)-£  +  sec  ft  sin  c  sin  (  C  +  w)  -j- 

dt  at 

-|-(cos  c  —  tan  ft  sin  c  sin  (  C  +  to))  -?-, 

by  means  of  which  the  components  of  the  velocity  of  the  disturbed 
body  in  directions  parallel  to  the  co-ordinato  axes  may  be  determined. 

The  values  of  -^  and  -^  will  be  obtained  from  -^  and  -^  by  a 
single  integration,  and  then  we  have 


474  THEORETICAL   ASTRONOMY. 

dw       &V/pT(l-hm)    ,    d$w  dr        k\/l-\-m 


, 

I  ~    -     -3P          I        -^T°°     -di'   (85) 

from  which  to  find  —=-  and  —  :-• 
at  at 

175.  EXAMPLE.  —  In  order  to  illustrate  the  calculation  of  the  per- 
turbations of  r,  w,  and  z,  let  us  take  the  data  given  in  Art.  166,  and 
determine  these  perturbations  instead  of  those  of  the  rectangular  co- 
ordinates. 

In  the  first  place,  we  derive  from  the  tables  of  the  motion  of 
Jupiter  the  values 

£'  =  98°  58'  22".7,  i'  =  1°  18'  40".5, 

which  refer  to  the  ecliptic  and  mean  equinox  of  1860.0.  We  find, 
also,  from  the  data  given  by  the  tables  the  values  of  u'  measured 
from  the  ecliptic  of  1860.0.  Then,  by  means  of  the  formulae  (66), 
using  the  values  of  &0  and  i0  given  in  Art.  166,  we  derive 

N=  194°  0'  49".9,  N'  =  301°  38'  31".7,  1=  5°  9'  56".4. 

The  value  of  u0f  is  given  by  equation  (68),  and  then  wf  and  $'  are 
found  from  the  equations  (67).  Thus  we  have 

Berlin  Mean  Time.  log  r0  w0  =  UQ  log  r  wf  ft' 

1863  Dec.  12.0,  0.294084  192°  4'  24".5  0.73425  14°  18'  54".6  —  0°  V  38".l 

1864  Jan.  21.0,  0.294837  207  40  52  .2  0.73368  17  21  44  .2  0  18  9  .1 
March  1.0,  0.300674  223  3  5  .9  0.73305  20  25  5  .2  0  34  39  .9 
April  10.0,  0.310864  237  51  38  .3  0.73237  23  28  59  .8  0  51  7  .6 
May  20.0,  0.324298  251  52  47  .9  0.73164  26  33  32  .1  17  29  .7 
June  29.0,  0.339745  264  59  30  .0  0.73086  29  38  44  .8  1  23  43  .5 
Aug.  8.0,  0.356101  277  10  24  .6  0.73003  32  44  41  .2  1  39  46  .3 
Sept.  17.0,  0.372469  288  28  4  .1  0.72915  35  51  24  .6  1  55  35  .2 
Oct.  27.0,  0.388214  298  57  16  .3  0.72823  38  58  57  .5  2  11  7  .5 
Dec.  6.0,  0.402894  308  43  48  .7  0.72726  42  7  23  .3  2  26  20  .3 

1865  Jan.  15.0,  0.416240  317  53  39  .1  0.72625  45  16  43  .9  —2  41  10  .6 

The  values  of  p0  may  be  found  from  (76)  or  (78)  as  already  given  in 
Art.  166. 

The  forces  R,  S0,  and  Z  may  now  be  determined  by  means  of  the 
equations  (74),  h  being  found  from  (70),  and  if  we  introduce  the 
factor  (o2  for  convenience  in  the  integration,  as  already  explained,  we 
obtain  the  following  results  : 


Date.  6>2.K  u*S0r0  u*Z 

1863  Dec.  12.0,      +  1".4608       +  0".1476        4  0".0009       +  0".0282 

1864  Jan.  21.0.   +  1  .4223   —  0  .6757   4-  0  .0101   -  0  .2361 


NUMERICAL   EXAMPLE.  475 


Date. 

«IR 

rfS0r0 

tfZ 

u\  S0r0dt 

1864  March  1.0, 

+  1".2616 

—  1".4512 

-}•  0".0190 

—  1".3060 

April  10.0, 

1  .0018 

2  .1226 

0  .0273 

3  .1035 

May  20.0, 

0  .6760 

2  .6473 

0  .0347 

5  .5020 

June  29.0, 

+  0  .3179 

2  .9988 

0  .0406 

8  .3402 

Aug.   8.0, 

—  0  .0452 

3  .1650 

0  .0449 

11  .4378 

Sept.  17.0, 

0  .3944 

3  .1437 

0  .0470 

14  .6076 

Oct.  27.0, 

0  .7180 

2  .9392 

0  .0466 

17  .6640 

Dec.      6.0,  1  .0097  2  .5586  0  .0432  20  .4273 

1865  Jan.     15.0,      —  1  .2674       -  2  .0081      +  0  .0362      —  22  .7245 

The  integral  wl  SQr0dt  is  obtained  from  the  successive  values  of  Q)*SQr9 

by  means  of  the  formula  (32). 

Next  we  compute  the  values  of  the  differential  coefficients  by 
means  of  the  formulae  (65).  For  the  dates  1863  Dec.  12.0  and  1864 
Jan.  21.0  we  may  first  assume  dr  =  Q,  and,  by  a  preliminary  inte- 
gration, having  thus  derived  very  approximate  values  of  dr  for  these 

dates,  the  values  of  —^-  will  be  recomputed.     Then,  commencing 

anew  the  table  of  integration,  we  may  at  once  derive  an  approximate 
value  of  dr  for  the  date  March  1 .0  with  which  the  last  term  of  the 

expression  for  —7—  may  be  computed.  Continuing  this  indirect  pro- 
cess, as  already  illustrated  in  the  case  of  the  perturbations  of  the  rec- 
tangular co-ordinates,  we  obtain  the  required  values  of  the  second 

differential  coefficient.  In  a  similar  manner,  the  values  of  -TZ  will 
be  obtained.  The  values  of  —=r  will  then  be  given  directly  by  means 

of  the  first  of  equations  (65) ;  and  the  final  integration  will  furnish 
the  perturbations  required.  Thus  we  derive  the  following  results  :- 

1863  Dec.  12.0,  —  0".0423  -j-l".4509  +0".0009  —  0".00  +  0".18  +0".00 

1864  Jan.  21.0,  0  .1086  1  .3405  0  .0101  0  .02  0  .17  0  .00 
Mar.  1.0,  0  .7162  +0  .7829  0  .0183  0  .40  1  .47  0  .01 
Apr.  10.0,  1.6114—0.0455  0.0251  1.55  3.53  0.04 
May  20.0,  2  .4795  0  .9344  0  .0300  3  .61  5  .54  0  .09 
June  29.0,  3  .0807  1  .7333  0  .0326  6  .42  6  .62  0  .18 
Aug.  8.0,  3  .2971   2  .3752  0  .0331  9  .64  5  .98  0  .29 
Sept.  17.0,  3  .1080  2  .8533  0  .0311  12  .88  +2  .98  0  .44 
Oct.  27.0,  —2  .5425  —3  .1872  4-0  .0265  —15  .73  —2  86  +0  .62 


476  THEORETICAL   ASTRONOMY. 


1864  Dec.    6.0,  —  1".6443  —  3".4009  -f  0".0190  —  17".85  —  11".88  +0".83 

1865  Jan.  15.0,  —0  .4511—3  .5334  -fO  .0079—18  .92—24  .29+1  .05 

It  has  already  been  found  that,  during  the  period  included  by  these 
results,  the  perturbations  arising  from  the  squares  and  products  of 
the  disturbing  forces  are  insensible,  and  hence  the  application  of  the 
complete  equations  for  the  forces  and  for  the  differential  coefficients 
is  not  required.  The  equations  (83)  will  give,  by  means  of  the 
results  for  w  —  u0  -f-  dw,  r  =  r0  -f-  dr,  and  z,  the  values  of  the  helio- 
centric co-ordinates  of  the  disturbed  body,  and  the  combination  of 
these  with  the  co-ordinates  of  the  sun  will  give  the  geocentric  place. 
When  we  neglect  terms  of  the  second  order,  we  have,  according  to 
the  equations  (84), 

y 

dxlf  =  XQ  cot  (  A  -j-  w]  dw  -f  _  °  dr  -f  z  cos  a, 

ro 

dy,,  =  y0  cot  (B  -f  w)  dw  -f  ??  dr  -f  z  cos  b,  (86) 

ro 

dzn  =  z0  cot  (C  -{-  w)  dw  -{-  —  $r  -\-  z  cos  c, 

ro 

the  heliocentric  co-ordinates  x0)  y0)  ZQ  being  referred  to  the  same  fun- 
damental plane  as  the  auxiliary  constants,  a,  6,  A,  &c.  Thus,  in  the 
case  of  Eurynome,  to  find  the  perturbations  of  the  rectangular  co-or- 
dinates, referred  to  the  ecliptic  and  mean  equinox  of  1860.0,  from 
1864  Jan.  1.0  to  1865  Jan.  15.0,  we  have 

A  =  296°  34X  37x/.5,  B  =  206°  437  34".4,          C=Q, 

log  cos  a  =  8.557354n,          log  cos  b  =  8.856746,          log  cos  c  =  log  cos  i0  =  9.998590, 
log  XQ  =  0.399807n,  log  y0  =  9.838709,  log  z0  =  9.148170,, 

w  =  w0  +  6W  =  317°  53r  20//.2, 

and  hence,  by  means  of  (86),  we  derive 

3x,  =  +  36".559,        fy,  =  +  41".083,        dz,  =  —  0".588. 

If  we  express  these  in  parts  of  the  unit  of  space,  and  in  units  of  the 
seventh  decimal  place,  we  obtain 

dx,  =  +  1772.4,        dy,  =  +  1991.8,        to,  =  —  28.5, 

agreeing  with  the  results  already  obtained  by  the  method  of  the  va- 
riation of  rectangular  co-ordinates,  namely, 

9x,  =  +  1772.6,        fy,  =  f  1992.3,        to,  =  —  28.2. 


CHANGE   OF   THE   OSCULATING   ELEMENTS.  477 

176.  By  using  the  complete  formulae,  the  perturbations  of  r,  w, 
and  z  may  be  computed  with  respect  to  all  powers  of  the  disturbing 
force,  and  for  a  long  series  of  years,  using  constantly  the  same  fun- 
damental osculating  elements.  But  even  when  these  elements  are  so 
accurate  as  not  to  require  correction,  on  account  of  the  effect  of  the 
large  perturbations  of  long  period  upon  the  values  of  dw  and  dr,  the 
numerical  values  of  the  perturbations  will  at  length  be  such  that  a 
change  of  the  osculating  elements  becomes  desirable,  so  that  the 
integration  may  again  commence  with  the  value  zero  for  the  variation 
of  each  of  the  co-ordinates.  This  change  from  one  system  of  ele- 
ments to  another  system  may  be  readily  effected  when  the  values  of 
the  perturbations  are  known.  Thus,  having  found  the  disturbed 
values  of  r,  w,  and  z,  we  have 

dv*  .-did*   ,   dp*      Vp(l  +  ro) 

_  =  cos./?_  +  _==__  --  , 

p  being  the  semi-parameter  of  the  instantaneous  orbit  of  the  disturbed 
body.     In  the  undisturbed  orbit  we  have 


_  dv0 
ffo~  dt 
and  hence  we  derive 


dv* 


Substituting  for  -5-  the  value  above  given,  there  results 

1     ddw 


„„ 


J  n 

by  means  of  which  p  may  be  determined.     To  find  --^»  we  have 
d3  1         dz       ian/3    dr 


~dt        rcos/5    ~df          r        dt 
We  have,  also, 

dr       kVT+^n,      .  &l/l-f-m      .  dfr 

—  -  •_  e  sm  v  =  --  /—  —  e0  sin  v0  +  -^-, 
dt  i/p  VpQ 

and  if  we  put 

(89) 

(    ; 


p, 


478  THEORETICAL   ASTRONOMY. 

this  equation  becomes 

e  sin  v  =  en  sin  v0  -f-  aeQ  sin  v0  -f-  Y.  (90) 

We  have,  further. 

ecosv  =  -  —  1, 
r 

and,  putting 

*•?-!+*  («) 

we  obtain 

e  cos  r  =  e0  cos  v0  -}-  /5  —  . 
ro 

This  equation,  combined  with  (90),  gives 

e  sin  (v  —  v0)  =  aen  sin  v0  cos  v0  -f-  f  cos  v0  —  —  $  sin  v0, 

p  (92) 

e  cos  (v  —  v0)  =  e0  -f-  ae0  sin2  v0  +  r  sin  v0  +  -  ft  cos  v0, 

ro 

by  means  of  which  the  values  of  e  and  v  may  be  found,  those  of  the 
auxiliaries  a,  ft  ft  being  found  from  (89)  and  (91).     Then  we  have 

e  =  sin  <f>,  a=p  sec*  <f>, 

p  =  ^1/1+m,  tan  $E  =  tan  (45°  —  £?)  tan  |v, 

a^ 

M=E  —  esinE, 

by  means  of  which  ^,  a,  //,  and  Jf  may  be  determined.     In  the  case 
of  orbits  of  great  eccentricity,  we  find  the  perihelion  distance  from 


and  the  time  of  perihelion  passage  will  be  derived  from  e  and  v  by 
means  of  Table  IX.  or  Table  X. 

It  remains  yet  to  determine  the  values  of  £2,  i,  and  a>  or  it.  Let 
6C  denote  the  longitude  of  the  ascending  node  of  the  instantaneous 
orbit  on  the  plane  of  the  osculating  orbit,  defined  by  &0  and  iQ)  mea- 
sured from  the  origin  of  w,  and  let  ^0  denote  its  inclination  to  this 
plane.  Then  we  have 

tan  fj0  sin  (w  —  00)       =  tan  /?, 

.  ,  dw  Odp  (93) 

tan  i?0  cos  (w  —  0^  —  =  sec'  /?  -^, 

and  hence 


CHANGE   OF   THE   OSCULATING   ELEMENTS.  479 

.dfiw 

g»  +  ~df 
tan  (w  —  O  =  ^sin  2/9      ^      ,  (94) 

"dT 

by  means  of  which  00  may  be  found.  The  quadrant  in  which  00  is 
situated  is  determined  by  the  condition  that  sin  (w  —  00)  and  tan  /3 
must  have  the  same  sign.  The  value  of  %  will  be  found  from  the 
first  or  the  second  of  equations  (93). 

If  we  denote  by  £  the  argument  of  the  latitude  of  the  disturbed 
body  with  respect  to  the  adopted  fundamental  plane,  we  have 


cos>?0 


(95) 


and  the  angle  £  must  be  taken  in  the  same  quadrant  as  w  —  00. 
Then,  from  the  spherical  triangle  formed  by  the  intersection  of  the 
planes  of  the  ecliptic  and  instantaneous  orbit  of  the  disturbed  body, 
and  the  fundamental  plane,  with  the  celestial  vault,  we  derive 

cos  \  i  sin  (£  (u  —  C)  -f-  A  (&  —  &0))  =  sin  J<?0  cos  \  (i0  —  T?O), 
cos  ^  i  cos  Q  (u  —  C)  -f-  £  (&  —  &<>))  —  cos  2^0 cos  i  (*o  ~f~  ^o); 
sin  ^  i  sin  (^(u  —  C)  —  i  ( &  —  &„))  =  sin  A00  sin  ^  (t0  —  ^0), 
sin  \  i  cos  {^(u  —  C)  —  ^  ( &  —  &0))  :r=  cos  l^o  s^n  i  (*o  ~f~  7o)' 

These  equations  will  furnish  the  values  of  i,  u  —  f ,  and  ^  —  £ 
hence,  since  £  and  &0  are  given,  those  of  R>  and  u.     The  value  of  v 
having  been  already  found,  we  have,  finally, 

a)=U  —  V, 


and  the  elements  are  completely  determined.  These  elements  will 
be  referred  to  the  ecliptic  and  mean  equinox  to  which  &0  and  iQ  are 
referred,  and  they  may  be  reduced  to  the  equinox  and  ecliptic  of  any 
other  date  by  means  of  the  formulae  which  have  already  been  given. 
The  elements  of  the  instantaneous  orbit  of  the  disturbed  body  may 
also  be  determined  by  first  computing  the  values  of  #,„  y,,,  z,n  in 
reference  to  the  fundamental  plane  to  which  &  and  i  are  to  be  re- 
ferred, by  means  of  the  equations  (83),  and  also  those  of  -|p  -,'-',  -~ 

by  means  of  (85)  and  (84),  and  then  determining  the  elements  from 
the  co-ordinates  and  velocities,  as  already  explained. 

It  should  be  observed  that  when  the  factor  w2,  or  the  square  of  the 


480  THEORETICAL   ASTRONOMY. 

adopted  interval,  is  introduced  into  the  expressions  for  the  forces  and 
differential  coefficients,  the  first  integrals  will  be 

dSr  dtiw  dz 

••*•      "IT       "dp 

and  that  when  these  quantities  are  expressed  in  seconds  of  arc,  they 
must  be  converted  into  their  values  in  parts  of  the  unit  of  space 
whenever  they  are  to  be  combined  with  quantities  which  are  not  ex- 
pressed in  seconds.  In  other  words,  the  homogeneity  of  the  several 
terms  must  be  carefully  attended  to  in  the  actual  application  of  the 
formulae. 

When  the  elements  which  correspond  to  given  values  of  the  per- 
turbations have  been  determined,  if  we  compute  the  heliocentric 
longitude  and  latitude  of  the  body  for  the  instant  to  which  the  ele 
ments  belong,  the  results  should  agree  with  those  obtained  by  com- 
puting the  heliocentric  place  from  the  fundamental  osculating  ele- 
ments and  adding  the  perturbations. 

177.  The  computation  of  the  indirect  terms  when  the  perturba- 
tions of  the  co-ordinates  r,  w,  and  z  are  determined,  is  effected  with 
greater  facility  than  in  the  case  of  the  rectangular  co-ordinates, 
although  the  final  results  are  not  so  convenient  for  the  calculation  of 
an  ephemeris  for  the  comparison  of  observations.  This  indirect  cal- 
culation, which,  when  the  perturbations  of  any  system  of  three  co- 
ordinates are  to  be  computed,  cannot  in  any  case  be  avoided  without 
impairing  the  accuracy  of  the  results,  may  be  further  simplified  by 
determining,  in  a  peculiar  form,  the  perturbations  of  the  mean 
anomaly,  the  radius-vector,  and  the  co-ordinate  z  perpendicular  to  the 
fundamental  plane  adopted. 

Let  the  motion  of  the  disturbed  body  be,  at  each  instant,  referred 
to  the  plane  of  its  instantaneous  orbit;  then  we  shall  have  /9  =  0, 
and  the  equations  (51)  and  (54)  become 

T>  ^  =  far  dt  +  kl/p^l  +  m), 

at      J  ,0- 

d?r         dw*   ,    &'(!  +m)  _ 
dt*       T  dt3  H  r*  ' 

in  which  R  denotes  the  component  of  the  disturbing  force  in  the 
direction  of  the  disturbed  radius- vector,  and  S  the  component  in  the 
plane  of  the  disturbed  orbit  and  perpendicular  to  the  disturbed  radius- 
vector,  being  positive  in  the  direction  of  the  motion.  The  effect  of 


VARIATION   OF   POLAR   CO-ORDINATES.  481 

the  components  R  and  S  is  to  vary  the  form  of  the  orbit  and  the 
angular  distance  of  the  perihelion  from  the  node.  If  we  denote  by 
Z  the  component  of  the  disturbing  force  perpendicular  to  the  plane 
of  the  instantaneous  orbit,  the  eifect  of  this  will  be  to  change  the 
position  of  the  plane  of  the  orbit,  and  hence  to  vary  the  elements 
which  depend  on  the  position  of  this  plane. 

Let  us  take  a  fixed  line  in  the  plane  of  the  instantaneous  orbit, 
and  suppose  it  to  be  directed  from  the  centre  of  the  sun  to  a  point 
whose  angular  distance  back  from  the  place  of  the  ascending  node  is 
0,  and  let  the  value  of  a  be  so  taken  that,  so  long  as  the  position  of 
the  plane  of  the  orbit  is  unchanged,  we  shall  have 


The  line  thus  taken  in  the  plane  of  the  orbit  may  be  regarded  as 
fixed  during  all  changes  in  the  position  of  this  plane.  Let  £  denote 
the  angle  between  this  fixed  line  and  the  semi-transverse  axis  ;  then 
will 

*  =  «  +  ',  (98) 

and  when  the  position  of  the  plane  of  the  orbit  is  unchanged,  we  have 


But  if,  on  account  of  the  action  of  the  component  Z,  the  position  of 
the  plane  of  the  orbit  is  changed,  we  have,  according  to  the  equations 
(72)2,  the  relations 


&,  (99) 

dn  =dz  +  (l  —  wai)dtt  =dfc  +  28inflida. 

We  have,  further, 


v  being  the  true  anomaly  in  the  instantaneous  orbit. 

The  two  components  of  the  disturbing  force  which  act  in  the  plane 
of  the  disturbed  orbit  will  only  vary  /  and  the  elements  which  deter- 
mine the  dimensions  of  the  conic  section.  We  have,  therefore,  in  the 
case  of  the  osculating  elements,  for  the  instant  t0, 


Let  us  now  suppose  /I  to  denote  the  true  longitude  in  the  orbit,  so 

that  we  have 

J  =  t;-f7r  =  v+a>-t-a, 
31 


482  THEORETICAL   ASTRONOMY. 

or 

*  =  «+*  —  ('—£);  (101) 

then,  since  £  is  equal  to  TZ  when  the  position  of  the  plane  of  the  orbit 
is  unchanged,  it  follows  that  a  —  &  represents  the  variation  of  the 
true  longitude  in  the  orbit  arising  from  the  action  of  the  component 
Z  of  the  disturbing  force.  The  elements  may  refer  to  the  ecliptic  or 
the  equator,  or  to  any  other  fundamental  plane  which  may  be  adopted. 

178.  For  the  instant  t  we  have,  in  the  case  of  the  disturbed  motion, 
the  following  relations  :  — 

E—  e  sin  E=  M  +  ft  (t  —  y, 

r  cos  v  =  a  cosE  —  ae,  fl  02") 

r  sin  v  =  al/1  —  e2  sin  E, 

*=*+*—  o—  a). 

Let  us  first  consider  only  the  perturbations  arising  from  the  action  of 
the  two  components  of  the  disturbing  force  in  the  plane  of  the  dis- 
turbed orbit,  and  let  us  put 

(103) 


Further,  let  Jf0  +  /*0  (t  —  Q  -f-  8M  be  the  mean  anomaly  which,  by 
means  of  a  system  of  equations  identical  in  form  with  the  preceding, 
but  in  which  the  values  of  a0,  eQ,  £0  are  used  instead  of  the  instanta- 
neous values  a,  6,  and  £,  gives  the  same  longitude  ^,,  so  that  we  have 


r,  cos  v,  =  a0  cos  E,  —  a0e0, 
r,  sin  v,  =  a0T/l  —  e02  sin  Ef 


If,  therefore,  we  determine  the  value  of  dM  so  as  to  satisfy  the  con- 
dition that  ^,  =  v  +  £,  the  disturbed  value  of  the  true  longitude  in 
the  orbit,  neglecting  the  effect  of  the  component  ^of  the  disturbing 
force,  will  be  known.  The  value  of  r,  will  generally  differ  from  that 
of  the  disturbed  radius-vector  r,  and  hence  it  becomes  necessary  to 
introduce  another  variable  in  order  to  consider  completely  the  effect 
of  the  components  R  and  S.  Thus,  we  may  put 

r  =  r,(l  +  0,  (105) 

and  v  will  always  be  a  very  small  quantity.  When  dM  and  v  have 
been  found,  the  effect  of  the  disturbing  force  perpendicular  to  the 
plane  of  the  instantaneous  orbit  may  be  considered,  and  thus  the 
Complete  perturbations  will  be  obtained. 


VARIATION   OF   CO-ORDINATES.  483 

In  the  equations  (97),  ^~JT  expresses  the  areal  velocity  in  the  in- 

stantaneous orbit,  and  it  is  evident  that,  since  the  true  anomaly  is  not 
affected  by  the  force  ^perpendicular  to  the  plane  ol  the  actual  orbit, 

\r^  -jj-  must  also  represent  this  areal  velocity,  and  hence  the  equations 
become 

^  =  fSrdt 


, 
W    r\~dt}^       ~^~ 

179.  If  we  differentiate  each  of  the  equations  (104),  we  get 


dr.  .       dv,  .    ,.,  dE, 

cos  v,  -£  —  r,smv,-j-  =  —  aa  sm  E,  -=t 

*  *  '  (107) 


sin  v,  -j-  +  r,  cos  v,  -£  =  a0T/l — e0*  cos  .E,  —=--', 

(it  dt  dt 

~dt~'=~di' 
From  the  second  and  the  third  of  these  equations  we  easily  derive 

cosE-ar  cos*  sm£)— '. 
0  '  '    dt 

it  77* 

Substituting  in  this  the  values  of  r,  sin  vn  r,  cos  v,,  and  ^p  and  re- 
ducing, we  get 

rdrL_ 

or 

W  = 

•    JTV 

dr, 
From  the  same  equations,  eliminating  — ,  we  get 

r,» -^r  =  (a0l^l  —  e*r>  cos vt  cos ^  +  aor^ sm  v/ sm -^/)  -gj-'» 
which  reduces  to 

+  -~i  (109) 


.       /-    ,    1     ^Jf\  /1AQ>. 

m  v,    1  H ^T-  .  (108) 

\         HQ      dt   1 


484  THEORETICAL   ASTRONOMY. 


Combining  this  with  the  first  of  equations  (106),  we  get 


from  which  dM  may  be  found  as  soon  as  v  is  known. 
The  equation  (105)  gives 

dr  dr,  dv 

—  =  (l+v)^  +  2^.  —  +  r— . 

(Li) 

Differentiating    equation    (108)   and   substituting  for  -^  its  value 
already  found,  we  obtain 

~d&=      ~~^     \+v0'~dr)  + 

and  the  last  of  the  preceding  equations  becomes 

^  =  r,j£  H ™ ><  e0cosv,  ^  +  ^ 


.  .  ,   0  rfv        2 

___eoSm__,__  +  2__  +  - 


The  equation  (110)  gives 

2       dv         2 


s  ,  ,. 

" 


which  is  easily  reduced  to 

~fT0         d?~^~<*~dt  ~^v~0'  ~dt'~dT'=l  +  v* 
and  hence  we  derive 


The  equation  (109)  gives 


VARIATION   OF   CO-ORDINATES.  485 

dv,\*_  ffip0(l  +  m)    I  1     d3M\* 

and,  since 


this  becomes 


,  m    3,0081),,, 

~~ 


,     (H3) 


Combining  equations  (112)  and  (113)  with  the  second  of  equations 
(106),  we  get 

^v_l  +  vp   , 
=    "-*- 


From  (110)  we  derive 


and  the  preceding  equation  becomes 


0  ^ 

or  at 


which  is  the  complete  expression  for  the  determination  of  v. 

180.  It  remains  now  to  consider  the  effect  of  the  component  of  the 
disturbing  force  which  is  perpendicular  to  the  plane  of  the  disturbed 
orbit.  Let  xn  yn  zf  denote  the  co-ordinates  of  the  body  referred  to 
the  fundamental  plane  to  which  the  elements  belong,  and  x,  y  the 
co-ordinates  in  the  plane  of  the  instantaneous  orbit.  Further,  let  a 
denote  the  cosine  of  the  angle  which  the  axis  of  x  makes  with  that 
of  xn  and  /9  the  cosine  of  the  angle  which  the  axis  of  y  makes  with 
that  of  y,,  and  we  shall  have 

z,  =  ax  +  {3y.  (116) 

If  the  position  of  the  plane  of  the  orbit  remained  unchanged,  these 


486  THEORETICAL  ASTRONOMY. 

cosines  a  and  /9  would  be  constant;  but  on  account  of  the  action  of 
the  force  perpendicular  to  the  plane  of  the  orbit,  these  quantities  are 
functions  of  the  time.  Now,  the  co-ordinate  z,  is  subject  to  two  dis- 
tinct variations  :  if  the  elements  remain  constant,  it  varies  with  the 
time;  and,  in  the  case  of  the  disturbed  orbit,  it  is  also  subject  to  a 
variation  arising  from  the  change  of  the  elements  themselves.  We 
shall,  therefore,  have 


dt  ~\  dt 
in  which  I  -^  1  expresses  the  velocity  resulting  from  the  constant 

elements,  and      -£      that  part  of  the  actual  velocity  which  is  due 

to  the  change  of  the  elements  by  the  action  of  the  disturbing  force. 
But  during  the  element  of  time  dt  the  elements  may  be  regarded  as 

constant,  and  hence  the  velocity  -~  in  a  direction  parallel  to  the 

axis  of  zf  may  be  regarded  as  constant  during  the  same  time,  and  as 
receiving  an  increment  only  at  the  end  of  this  instant.  Hence  we 
shall  have 


d*L_(dzL\ 
dt~\dt] 


Differentiating   equation  (116),  regarding  a  and  /9  as  constant,  we 

get 

dz,  \      dz,  dx          dy 


and  differentiating  the  same  equation,  regarding  x  and  y  as  constant, 
we  get 

(118) 


Differentiating  equation  (117),  regarding  all  the  quantities  involved 
as  variable,  the  result  is 

d*z,_da    dx       d?     dy         d*x         <Py  Q19) 

W"~dt   ~dt^~~di  ~dt     *M+*dF 

Now,  we  have 

Z,  =  aX+  0Y+  Zcosi,  (120) 


in  which  Z,  denotes  the  component  of  the  disturbing  force  parallel 
to  the  axis  of  z,,  and  i  the  inclination  of  the  instantaneous  orbit  to 


VARIATION   OF   CO-OEDINATES.  487 

the  fundamental  plane.    Substituting  for  X  and  Y  their  values  given 
by  the  equations  (1),  and  reducing  by  means  of  (116),  w*  obtain 


or 


Comparing  this  with  (119),  there  results 
da,     dx    .    dft     dy 

TB-sf  +  TB-aH 

181.  The  equation  (120)  gives 

r.       (122J 


The  component  of  the  disturbing  force  perpendicular  to  the  plane  of 
the  disturbed  orbit  does  not  affect  the  radius-  vector  r  ;  and  hence, 
when  we  neglect  the  effect  of  this  component,  and  consider  only  the 
components  R  and  S  which  act  in  the  plane  of  the  orbit,  we  have 


d\ 

dp   ~~       —  ^3  --  zo  +  ao^-  T 

in  which  z0  denotes  the  value  of  z,  obtained  when  we  put  Z=0. 
Let  us  now  denote  by  dz,  that  part  of  the  change  in  the  value  of  z, 
which  arises  from  the  action  of  the  force  perpendicular  to  the  plane 
of  the  disturbed  orbit,  so  that  we  shall  have 


Substituting  these  in  equation  (122)  and  then  subtracting  equation 
(123)  from  the  result,  we  get 

(124) 


) 

The  equations  (116)  and  (117)  give 

*-*  +  ,*  ^*fc  +  $* 

If  we  eliminate  5/9  between  these  equations,  there  results 
dy          dx  \       dy  ddz, 

---- 


488  THEORETICAL   ASTRONOMY. 

and  since  the  factor  of  da  in  this  equation  is  double  the  areal  velocity 
in  the  disturbed  orbit,  we  have 


"25> 


Eliminating  da  from  the  same  equations,  we  obtain,  in  a  similar 
manner, 

(126) 


Substituting  these  values  in  equation  (124),  it  becomes 


.   _  1  ll^dy        vdx\,         ,„ 

H  --  /  I    X-£  —  Y-rr  ]  dz,  -f-  (Fa;  — 

A;l/l      m\       dt  dt] 


. 

dt 

If  we  introduce  the  components  R  and  S  of  the  disturbing  force,  we 
have 


r          r 
and  hence 


r 

Yx     —Xy     =Sr. 

Therefore  the  equation  (127)  becomes 
tffc,  &2(l+mV 


dr\  Sr 


p  5  dr\ 

\  r       jfei/p(l+m)  '  *  /  Z' 

We  have,  further, 


which,  by  means  of  the  equations  (108)  and  (109),  gives 
dr          e0sinv,      9dv,    .        dv        ^ 


Substituting  this  value  in  the  equation  (128),  we  obtain 


VARIATION   OF   CO-ORDINATES.  489 


Sr  Iddz,          3z,  (129) 

\   ^  "    1  +  '  ' 


which  is  the  complete  expression  for  the  determination  of  dzt. 

182.  The  equations  (110),  (115),  and  (129)  determine  the  complete 
perturbations  of  the  disturbed  body.  The  value  of  v  must  first  bo 
obtained  by  an  indirect  process  from  the  equation  (115),  and  then  dM 
is  given  directly  by  means  of  (110).  The  value  of  dz  will  also  be 
Determined  by  an  indirect  process  by  means  of  (129). 

In  order  to  obtain  the  expressions  for  the  forces  It,  S,  and  Z,  let  w 
denote  the  longitude  of  the  disturbed  body  measured  in  the  plane  of 
the  instantaneous  orbit  from  its  ascending  node  on  the  fundamental 
plane  to  which  &  and  i  are  referred,  it  being  the  argument  of  the 
latitude  in  the  case  of  the  disturbed  motion.  Let  wf  denote  the  lon- 
gitude of  the  disturbing  body  measured  from  the  same  origin  and  in 
the  plane  of  the  orbit  of  the  disturbed  body,  and  let  $'  denote  its 
latitude  in  reference  to  this  plane.  Finally,  let  N,  N',  /,  and  u0' 
have  the  same  signification  in  reference  to  the  plane  of  the  instanta- 
neous orbit  that  they  have  in  reference  to  the  plane  of  the  undisturbed 
orbit  in  the  case  of  the  equations  (66).  Then  we  shall  have 


=  cos  --        sn      *  — 

^Jsini  (N—  N')  =  sin  J  (ft'— 


from  which  to  determine  N,  Nf,  and  /.     We  have,  also, 


tan  (wr  —  N)  —  tan  u9'  cos  I,  (131) 

tan  p  =  tan  /sin  (uf  —  JV), 

from  which  to  find  wf  and  /?',  uf  being  the  argument  of  the  latitude 
of  the  disturbing  body  in  reference  to  the  plane  to  which  &  and  i 
are  referred. 

Since,  when  the  motion  of  the  disturbed  body  is  referred  to  the 
plane  of  its  instantaneous  orbit,  /?  =  0,  the  equations  (71),  (72),  and 
(73)  become 

R  =  m'telhi*  cosfl  cos(wf  —  w)  —  -t  }, 

*  p  '  (132) 

8  =  m'tfh  r'  cos  ft  sin  (wf  —  w), 

Z  =m'k2hrf  sin/5', 


490  THEORETICAL   ASTRONOMY. 

by  means  of  which  the  required  components  of  the  disturbing  force 
may  be  found,  the  value  of  h  being  given  by 


To  find  p,  we  have 

f  =  r'2  +  r8  —  2rr'  cos  p  cos  (yf  —  10),  (133) 

or,  putting 

cos  Y  —  cos  f?  cos  (wf  —  w), 
the  equations 

P  sin  w  =  /  sin  ?, 

p  cos  n  =  r  —  r'  cos  y. 

The  values  of  r'  and  u'  for  the  actual  places  of  the  disturbing 
body  will  be  given  by  the  tables  of  its  motion,  and  the  actual  values 
of  &  '  and  V  will  also  be  obtained  by  means  of  the  tables.  The  de- 
termination of  the  actual  values  of  r  and  w  requires  that  the  pertur- 
bations shall  be  known.  Thus,  when  dM  and  v  have  been  found, 
we  compute,  by  means  of  the  mean  anomaly  MQ  -f-  f20(t  —  t0)  +  dM 
and  the  elements  a0,  e0)  the  values  of  v,  and  r,.  Then,  since 
v  -f-  %  =  v,  -f-  TTO,  we  have,  according  to  (100), 

W  =  V,  +  7T0—  ff.  (135) 

We  have,  also, 


Tn  the  case  of  the  fundamental  osculating  elements,  we  have 


which  may  be  used  as  an  approximate  value  of  <r;  but  the  complete 
determination  of  w  requires  that  0=  &0  -f  8a  shall  also  be  deter- 
mined. The  exact  determination  of  the  forces  also  requires  that  the 
actual  values  of  Q>  and  i  as  well  as  those  of  Q,  '  and  i',  shall  be  used 
in  the  determination  of  N,  N',  and  I  for  each  instant.  When  these 
have  been  found,  it  will  be  sufficient  to  compute  the  actual  values  of 
N,  N'j  and  Jat  intervals  during  the  entire  period  for  which  the  per- 
turbations are  required,  and  to  interpolate  their  values  for  the  inter- 
mediate dates.  The  variations  of  these  quantities  arising  from  the 
variations  of  Q  ,  i,  &  ',  and  if  may  also  be  determined  by  means  of 
differential  formulae.  Thus,  from  the  differential  relations  of  the 
parts  of  the  spherical  triangle  from  which  the  equations  (130)  aie 
derived,  we  easily  find 


VARIATION   OF   CO-ORDINATES.  491 

,,T,       smi        ,          ^.,       ~x       sin  .AT  ,.        sinN 

dN  :=  - 


sini'        wjfnt       ~N       sinJV'    .,   ,    sinAr  (136) 


d/    =  cos  JV  di'  —  cos  -ZVdi  -}-  sin  £  sin  Nd  (ft'  —  ft  ). 
When  i  and  /  are  very  small,  it  will  be  better  to  use 

sin  i  sin  N'  sin  if  sin  JV 


sin/       sin(ft'— ft)'  sin/       sin(ft'  —  ft)' 


(137) 


in  finding  the  numerical  values  of  these  coefficients.  By  means  of 
these  formulae  we  may  derive  the  values  of  dN,  dN',  and  dl  corre- 
sponding to  given  values  of  £ft,  di,  £ft',  and  di1.  The  formulae 
by  means  of  which  da,  $ft,  and  di  may  be  obtained  directly,  will  be 
presently  considered. 

The  results  for  dN,  dN',  and  di  being  applied  to  the  quantities  to 
which  they  belong,  we  may  compute  the  actual  values  of  wr  and  /9'. 
The  value  of  r  will  be  found  from  the  given  value  of  v,  and  that  of 
w  will  be  given  by  means  of  equation  (135).  Then,  by  means  of 
the  formulae  (132),  the  forces  R,  S,  and  Z  will  be  obtained.  The 
perturbations  will  first  be  computed  in  reference  only  to  terms  de- 
pending on  the  first  power  of  the  disturbing  force,  and,  whenever  it 
becomes  necessary  to  consider  the  terms  of  the  second  order,  the 
results  already  obtained  will  enable  us  to  estimate  the  values  of  the 
perturbations  for  two  or  more  intervals  in  advance  with  sufficient 
accuracy  for  the  determination  of  the  three  required  components  of 
the  disturbing  force;  and  when  there  are  two  or  more  disturbing 
bodies  to  be  considered,  the  forces  for  each  of  these  may  be  computed 
at  once,  and  the  values  of  each  component  for  the  several  disturbing 
bodies  may  be  united  into  a  single  sum,  thus  using  2R,  2S,  and  2Z 
in  place  of  R,  S,  and  Z  respectively.  The  approximate  values  of  the 
perturbations  will  also  facilitate  the  indirect  calculation  in  the  deter- 
mination of  the  complete  values  of  the  required  differential  coeffi- 
cients. 

183.  When  only  the  perturbations  due  to  the  first  power  of  the 
disturbing  force  are  required,  the  osculating  elements  ft0  and  i0  will 
be  used  in  finding  N9  N',  and  /,  and  r0,  w0  will  be  used  instead  of  r 
and  w  in  the  calculation  of  the  values  of  R,  S,  and  Z.  The  equations 
for  the  determination  of  the  perturbations  dM,  v,  and  dz,9  neglecting 
terms  of  the  secotfi  order,  are,  according  to  the  equations  (110), 
(115),  and  (129),  the  following:— 


492  THEORETICAL   ASTRONOMY 

ddM  1 


frq+m), 

— -5 —  dz>- 


The  value  of  v  is  first  found  by  integration  from  the  results  given 
by  the  second  of  these  equations,  and  then  dM  is  found  from  the  first 
equation.  Finally,  dz,  is  found  by  means  of  the  last  equation.  The 
integrals  are  in  each  case  equal  to  zero  for  the  dates  to  which  the 
fundamental  osculating  elements  belong,  and  the  process  of  integra- 
tion is  analogous,  in  all  respects,  to  that  already  illustrated  in  the 

case  of  the  variation  of  the  rectangular  co-ordinates.     It  will  be  ob- 

d*v 
served,  however,  that  the  expression  for  -^-  involves  only  one  indi- 

rect term,  the  coefficient  of  which  is  small,  and  the  same  is  true  in 

.  d?dz,  ddM  . 

the  case  of  ~^r>  while  —  j^-  is  given  directly.     When  the  perturba- 

tions have  been  found  for  a  few  dates,  the  values  for  the  following 
date  can  be  estimated  so  closely  that  a  repetition  of  the  calculation 
will  rarely  or  never  be  required  ;  and  the  actual  value  of  r  may  be 
used  instead  of  the  approximate  value  r0  in  these  expressions  for  the 
differential  coefficients.  Neglecting  terms  of  the  second  order,  we 
have 

=  logr,  -f  V, 


wherein  ^0  denotes  the  modulus  of  the  system  of  logarithms.  We 
may  also  use  vf  instead  of  VQ;  but  in  this  case,  since  r,  and  v,  depend 
on  dM,  only  the  quantities  required  for  two  or  three  places  may  be 
computed  in  advance  of  the  integration. 

A  comparison  of  the  equations  (138)  with  the  complete  equations 
(110),  (115),  and  (129)  shows  that,  if  the  values  of  /3'  and  w'  are 
known  to  a  sufficient  degree  of  approximation,  we  may,  with  very 
little  additional  labor,  consider  the  terms  depending  on  the  squares 
and  higher  powers  of  the  masses.  It  will,  however,  appear  from 
what  follows,  that  when  we  consider  the  perturbations  due  to  the 
higher  powers  of  the  disturbing  forces,  the  consideration  of  the  effect 
of  the  variation  of  z,  in  the  determination  of  the  heliocentric  place 
of  the  disturbed  body,  becomes  much  more  difficult  than  when  the 
terms  of  the  second  order  are  neglected  ;  and  hence  it  will  be  found 
advisable  to  determine  new  osculating  elements  whenever  the  con- 
sideration of  these  terms  becomes  troublesome. 


VARIATION   OF   CO-ORDINATES.  493 

The  results  may  be  conveniently  expressed  in  seconds  of  arc,  and 
afterwards  v  and  Sz,  may  be  converted  into  their  values  expressed  in 
units  of  the  seventh  decimal  place,  or,  giving  proper  attention  to  the 
homogeneity  of  the  several  terms  of  the  equations,  in  the  numerical 
operations,  SM  may  be  expressed  in  seconds  of  arc,  while  v  and  8zf 
are  obtained  directly  in  units  of  the  seventh  decimal  place.  It  will 
be  advisable,  also,  to  introduce  the  interval  CD  into  the  formulse  in 
such  a  manner  that  this  quantity  may  be  omitted  in  the  case  of  the 
formulae  of  integration. 

184.  In  the  case  of  orbits  of  great  eccentricity,  the  mean  anomaly 
and  the  mean  daily  motion  cannot  be  conveniently  used  in  the  nu- 
merical application  of  the  formula?.  Instead  of  these  we  must 
employ  the  time  of  perihelion  passage  and  the  elements  q  and  e. 
Thus,  let  TQ  be  the  time  of  perihelion  passage  for  the  osculating  ele- 
ments for  the  date  t0,  and  let  T0  +  8T  be  the  time  of  perihelion  pas- 
sage to  be  used  in  the  formulae  in  the  place  of  T0  and  in  connection 
with  the  elements  q0  and  e0  in  the  determination  of  the  values  of  r, 
and  v,9  so  that  we  have 


In  the  case  of  parabolic  motion  we  have,  neglecting  the  mass  of  the 
disturbed  body, 

=s  (139) 

the  solution  of  which  to  find  v,  is  eifected  by  means  of  Table  VI.  as 
already  explained.  To  find  r,,  we  have 

r,  =  q0  sec2  %vt. 

For  the  other  cases  in  which  the  elements  MQ  and  fjt0  cannot  be  em- 
ployed, the  solution  must  be  effected  by  means  of  Table  IX.  or  Table 
X.  Thus,  when  Table  IX.  is  used,  we  compute  M  from 


wherein  log  <70  =  9.9601277,  and  with  this  as  the  argument  we  derive 
from  Table  VI.  the  corresponding  value  of  V.     Then,  having  found 

t  =  |rr-^  by  means  of  Table  IX.  we  derive  the  coefficients  required 

1  H~  en 

in  the  equation 

v,  =  V  +  A  (1000  +  B  (1000*  +  (7(1000',  (140) 


494  THEORETICAL    ASTRONOMY. 

from  which  v,  will  be  determined.     Finally,  r,  will  be  found  from 


When  Table  X.  is  used,  we  proceed  as  explained  in  Art.  41,  using 
the  elements  T—  T0  +  ST,  qw  and  ew  and  thus  we  obtain  the  required 
values  of  v,  and  r,. 

It  is  evident,  therefore,  that,  for  the  determination  of  the  pertur- 
bations, only  the  formula  for  finding  the  value  of  dM  requires  modi- 
fication in  the  case  of  orbits  of  great  eccentricity,  and  this  modifica- 
tion is  easily  effected.  The  expression 


gves 

^  <X  -  2;) 

or,  simply, 

dM 

and  the  equation  (110)  becomes 


(142) 

by  means  of  which  the  value  3T  required  in  the  solution  of  the  equa^ 
tions  for  r,  and  v,  may  be  found. 

If  we  denote  by  t,  the  time  for  which  the  true  anomaly  and  the 
radius-vector  computed  by  means  of  the  fundamental  osculating  ele- 
ments have  the  values  which  have  been  designated  by  v,  and  rty  re- 
spectively, we  have 


^-,  _._:   =_, 

and  the  equation  (110)  becomes 


(143) 

T";-      ^-r*J-    /cVp(t(i-\-m)*' 

or,  putting  t,  =  t-\-  dt, 

—  =  7 ^ 14-  7 r ==•    \  Sr  dt.       0  44) 

dt       (1  +  v)2  r  (1  +  v)2    k]/p0(l+m)  J 

If  we  determine  dt  by  means  of  this  equation,  the  values  of  the 
radius-vector  and  true  anomaly  will  be  found  for  the  time  t  +  dt 
instead  of  t,  according  to  the  methods  for  the  different  conic  sections, 


VARIATION   OF   CO-ORDINATES.  495 

using  the  fundamental  osculating  elements.  The  results  thus  obtained 
are  the  required  values  of  r,  and  v,  respectively. 

185.  When  the  values  of  the  perturbations  v,  Sz,,  and  dM,  8T,  or 
dt  have  been  determined,  it  remains  to  find  the  place  of  the  disturbed 
body.  The  heliocentric  longitude  and  latitude  will  be  given  by 

cos  bcos(l  —  ft  )  =  cos  (A  —  ft  ), 
cos  b  sin  (I  —  ft  )  =  sin  (A  —  ft  )  cos  t, 
sin  b  =  sin  (A  —  ft  )  sin  i, 

or,  since  X  =  X,  —  a  +  ft, 

cos  6  cos  (J  —  &  )  —  cos  (A,  —  <r), 

cos  6  sin  (I  —  ft  )  —  sin  (I,  —  <r)  cos  it  (145) 

sin  6  =  sin  (A,  —  (r)  sin  i, 

in  which  ^  =  9, -|- ftp  If  we  multiply  the  first  of  these  equations 
by  cos  (ft  — h),  and  the  second  by  — sin  (ft  — h\  in  which  h  may 
have  any  value  whatever,  and  add  the  results;  then  multiply  the  first 
by  (sin  ft  —  A),  and  the  second  by  cos  ( ft  —  h),  and  add,  we  get 

cos  b  cos  (I — Ji)=cos  (A, — <r)  cos  (  ft  — K)— sin  (A, — <r)  sin  (  ft  — A)  cos  i, 
cos  b  sin  (7 — Ji)=cos  (A, — «r)  sin  (  ft  — A)-f  sin  (A, — <r)  cos  (ft  — A)  cos  i, 
sin  6  ==fiin  (A, — <r)  sin  i. 

But,  since  A,  —  <r  =  (A,—  &0)  —  (<r—  Q0),  these  equations  may  be 
written 

cos  6  cos  (7  —  A) 

=cos  (A,— ^  0)  (cos  0—  &  0)  cos  (  £  —A)  +sin  (<r— ^  0)  sin  (  ^  —A)  cos  i) 
-f  sin  (A,— ft 0)  (sin  (>—  ft 0)  cos  (ft  —h)— cos  (>—  ft0)  sin  (ft  —  A)  cost), 

cos  6  sin  (Z  —  A)  (146) 

—cos  (A . — ft  0)  (cos  (<T—  ft  0)  sin  (  ft  —Ji) — sin  (<r—  ft  0)  cos  (  ft  —h)  cos  i 
-f  sin  (A,— ft0)(sin  (<r—  ft0)sin(ft—  h)+ cos(<r—  ft0)cos(ft—  K 

sin6=sin(Af—  ft0)cos(<r—  ft0)sini— cos  (A,— ft0)sin  (<r — ft0)  sin  i. 

Let  us  now  conceive  a  spherical  triangle  to  be  formed,  of  which  two 
of  the  sides  are  <r  —  ft0  and  ft  —A,  respectively,  and  let  the  angle 
included  by  these  sides  be  i.  Since  h  is  entirely  arbitrary,  we  may 
assign  to  it  a  value  such  that  the  other  angle  adjacent  to  the  side 
a  —  ft0  will  be  equal  to  i0.  Let  the  third  side  be  designated  by 
/i0—  ft0,  and  the  angle  opposite  to  a—  ft0  by  f]f.  The  auxiliary 
triangle  thus  formed  gives  the  following  relations : — 


496  THEORETICAL   ASTRONOMY. 


cos(A0—  0>o)=cos(ff—  &0)cos(&—  A)4-sin(<r—  &0)sin(&—  A)cosi, 
sin  (A0  —  &  0)  sin  i0  =sin  (  &  —  h)  sin  i,  (147) 

sin  (hQ  —  &  0)  cos  i0—  sin  (<r  —  &  0)  cos  (  &  —  A)  —  cos  (<r  —  Q  0)  sin  (  &  —  A)  cos  i, 
sin  (V-  &0)cosi/=cos(>—  &0)sin  (&  —  A)—  sin  (<r—  &0)cos(&  —  A)  cost. 

Combining  these  with  the  preceding  equations,  we  easily  derive 

cos  b  cos  (/—  A)=cos  (A,—  &0)  cos  (A0—  £0)-|-sin  0*,—  £0)  sin  (A0—  &0)  cosi0, 
cos&s:n  (/—A)—  sin  (>*,—  &0)cos(A0—  &0)  cos  t'0—  cos  (>*,—&  0)sin  (hQ—  &0) 
H-cos(/l,—  £0)sin(A0—  ^0)(l-f-cosV)  (148) 

4-sin  (A,  —  &0)  ((cos^  —  cosi0)  cos  (h0  —  ^0)-fsin  (<r—  £20)  sin  (Q  —  A)sin2^), 
8in&=sini0sin(A,  —  ^0)-}-(cos((r  —  ^0)sini  —  sin  i*0)  sin  (A,—  ^0) 

—  cos  (A,  —  £2  o)  sin  (ff  —  ^o)  s^u  i* 

Since  the  action  of  the  component  of  the  disturbing  force  perpen- 
dicular to  the  plane  of  the  disturbed  orbit  does  not  change  the  radius- 
vector,  we  have 

r  sin  b  =  r  sin  i0  sin  (V~&o)  H~  *>%„ 

and  hence  the  last  of  these  equations  gives 

-^  =  sin  (A,  —  £0)  (cos  (ff  —  &0)  sini  —  sin  i0) 

—cos  (J,  —  ^0)  sin  (<r  —  ^0)  sin  i. 

From  the  relation  of  the  parts  of  the  auxiliary  spherical  triangle,  we 

Have 

sin  i  sin  (<r  —  &0)  =  sin  if  sin  (A0  —  ^0), 

sin  i  cos  (<r  —  ^0)  =  sin  rf  cos  (hQ  —  ^0)  cos  iQ  +  cos  iy'  sin  i0. 


Therefore, 

A,  —  £  0)  (cos 

—  cos  (I,  —  &  0)  sin  (A0  —  &  0)  sin  i)', 


A  =  sin  (A,  —  £  0)  (cos  to  cos  (&„  —  ^0)  sin  if  —  sin  i0  (1  —  cos  if)),  ^ 


and 


1» *****  v    /        oo  u 
—  COST?  ,151) 

—  cos  (A, — &<>)sin(A0 — &0)  (l  +  cosi/)- 
We  have,  further,  from  the  auxiliary  spherical  triangle, 
cos  i  =  sin  i0  sin  >/  cos  (A0  —  ^  0)  —  cos  to  cos  if, 
from  which  we  get 

cos  i  —  cos  t'0  =  sin  t*0  cos  (A0  —  &  0)  sin  if  —  cos  t*0  (1  +  cos  if). 

We  have,  also, 

sin  (a  —  &  0)  sin  t = sin  rj  sin  ( AQ —  &  o)> 
sin  (  &  —  A)  sin  ?'  =  sin  i0  sin  (Ao —  &  0), 


VARIATION   OF   CO-ORDINATES.  497 

or 

sin  (<f  —  &o)  sin  (&  —  k)  sin2  i  =  sin3  (Ao  —  &0)  sin  t<>  sin  if. 

Hence  we  derive 

(cos  i  —  cos  i0)  cos  (h0  —  Q>  o)  -f  sin  (<r  —  &  0)  sin  (  &  —  h~)  sin2  i—  sin  i0  sin  V 

—  (1-j-cos  V)  cos  i0cos  (A0—  &0). 

Combining  this  and  the  equation  (151)  with  the  equations  (148),  we 
obtain 


cos  b  cos  (I  —  &)—  cos(A,  —  &0)  cos(/t0  —  &0)-fsin(A,  — 
cos  b  sin  (I  —  A)—  sin  (A,  —  &<>)  cos  (ho  —  &0)  cosi0  —  cos  (A,  —  &0)  sin  (h^  —  &„ 

sin  t{         Sz, 


sin  b  =sm  (A, — &  0)  sin  i^  -\ -. 

r 


1  —  COST/     r 


If  we  multiply  the  first  of  these  equations  by  cos(A0  —  &0),  and  the 
second  by  —  sm(/i0  —  &0),  and  add  the  results;  then  multiply  the 
first  by  sin  (h0  —  &0)>  and  the  second  by  cos  (h0  —  &0),  and  add,  we  get 


cosfc  cos(Z—  ^o—  (&—  Ao))=cos(A,—  ^0)+sin(^o—  S^o)  ~'  * 

* 


-~, 

T 


sin  b  =sin  (I, — &0)  sin  ^^ — '-•  (152) 

Let  us  now  put 

y=Sin(<r-a0)Smi, 


and  there  results,  from  (149), 

^-  =  qf  sin  (A,  -  fco)  -/  cos  ft  -  £0).  (154) 

Comparing  this  with  equation  (150),  we  observe  that 


pf  =  sn  >  sn     o  — 


c[  =  sin  V  cos  (&o  —  ^o)  cos  ^  —  sin  to  (1  —  cos  >?')• 
Therefore,  we  have 


cos  »  ~  o«>  =  tan  *• 

32 


498  THEORETICAL   ASTRONOMY. 

and,  if  we  put  F=  h  —  h0,  the  equations  (152)  became 
cos  b  cos  Mo_r)=cos  ft  -a  J+_J^_,  -  £ 
cosfcsin     --  -  - 


sin  6  =sin  (A,  —  &0)  sin  i0-]  —  -• 

As  soon  as  71,  p',  5',  and  ^'  are  known,  these  equations  will  furnish 
the  exact  values  of  I  and  6,  those  of  X,  and  r  being  found  by  means 
of  the  perturbations  v  and  dM. 

186.  The  value  of  F  may  be  expressed  in  terms  of  pf  and  <?'. 
Thus,  if  we  differentiate  the  first  of  equations  (147)  and  reduce  by 
means  of  the  remaining  equations  of  the  same  group,  we  get 

d(hQ  —  &0)  =  cosi/^(ft  —  ^)  ~f~  cosi0dff  -f-  smi0sm(<j-  —  &0)c?i, 


and  if  we  interchange  &  —  h  and  h0  —  &0  in  this  equation,  we  must 
also  interchange  i  and  i0,  which  are  the  angles  opposite  to  these  sides, 
respectively,  in  the  auxiliary  spherical  triangle,  so  that  we  shall  have 

d  (  &  —  A)  =  cos  if  d(h0  —  &0)  +  cos  i  d<r, 

i0  being  constant.  Adding  these  equations,  observing  that  &0  is  also 
constant,  we  get 

(1  —  COST/)  d(Q  —  A-fA)=sm^osm  (<r  —  &0)  c?i+(cosi4-cosi0)  dff',  (156) 
and  since  dff  =  cos  i  c?&,  this  becomes 

(1  —  cos  V)  d(h  —  h0)  =  —  sin  i0  sin  (<r  —  ^0)  di 

-f-  (sin2i  —  cos  V  —  cosi  cosi0)  -  :, 

COS  1 

which,  since 

cos  tj  =  sin  i  sin  i0  cos  (ff  —  $20)  —  cos  i  cos  i0,  (157) 

may  be  written 


(1  —  cos  i/)  dP=  —  sin  ^  sin  (<r  —  ^  0)  di+tsm  i  (sin  i  —  sin  i^  cos  (<r  —  &  „))  ^<r- 

(158) 
The  differentiation  of  the  equations  (153)  gives 

dp'  =  sin  (ff  —  &0)  cos  i  di  -)-  sin  i  cos  (ff  — 
dq'  =  cos  (ff  —  &0)  cos  i  di  —  sin  i  sin  (ff  — 

from  which  we  derive 


VARIATION   OF   CO-ORDINATES.  499 

q'dp  — p'dc[  =  sin2  i  d<r  —  sin  iQ  dp' 

=  cos  i  ( — sin  i0  sin  (<r — &o)  d i -\-tani  (sin  i — sin  i9  cos  (ff — &0))  d<r) . 


Combining  this  with  equation  (158),  we  get 

cos  i  (1  —  cos  V)  dr  =  q'dp'  —p'dq', 
and  hence 


r  [    dt       *    dt     j.  ,.,_„, 

=  J  cos^(l-cosV)^ 

the  integral  being  equal  to  zero  for  the  instant  to  which  the  funda- 
mental osculating  elements  belong.  It  is  evident  from  the  equations 
(153)  that  pf  and  qf  are  of  the  order  of  the  first  power  of  the  dis- 
turbing forces,  and  hence,  since  if  differs  but  little  from  180° — (fc+fy), 
it  follows  that,  so  long  as  i  is  not  very  large,  Pis  at  least  of  the 
second  order. 

The  last  of  equations  (145)  gives 

zt  =  r  sin  i  sin  A,  cos  ff  —  r  sin  i  cos  A,  sin  ff, 
and  since 

x  =  r  cos  A,,  y  =  r  sin  A,, 

this  becomes 

2,  =  —  x  sin  i  sin  ff  -f-  y  sin  i  cos  ff. 

Comparing  this  with  equation  (116),  it  appears  that 

a  =  —  sin  i  sin  ff,  ft  =  sin  i  cos  ff,  (160) 

and  hence,  by  means  of  (153),  we  derive 

p'  =  —  »  cos  &0  —  ft  sin  &0, 
qf  =  —  a  sin  &0  -f-  ft  cos  &0  —  sin  i0, 
and  also 

dp'  o    da,        .          dft 

™®>0~df~Sm®>0~dt' 


From  the  equations  (118)  and  (121),  observing  that 


we  derive,  by  elimination, 

da  _          r  sin  A,  cos  i   «  d£  __  r  cos  A;  cos  i   „, 

~~     ~  :  ""' 


500  THEOEETICAL   ASTRONOMY. 

Therefore  we  shall  have 

dp'  =  rcosi  sin  (A,  —  &0)  ^ 
*  ~        Al/pCl  +  m) 

r  COS  *  COS  (^'  —  &o) 


oy  means  of  which  pf  and  5'  may  be  found  by  integration,  the  inte- 
gral in  each  case  being  zero  for  the  date  t0  at  which  the  determina- 
tion of  the  perturbations  begins. 

When  the  value  of  dz,  has  already  been  found  by  means  of  the 
equation  (129),  if  we  compute  the  value  of  qf,  that  of  pf  will  be 
given  by  means  of  (154),  or 


and  if  pf  is  determined,  q'  will  be  given  by 

/  I  J.    f  1  /-V   \          I  > 


If  both  pf  and  q'  are  found  from  the  equations  (162),  dz,  may  be  de- 
termined directly  from  (154);  but  the  value  thus  obtained  will  be 
less  accurate  than  that  derived  by  means  of  equation  (129). 

Since  the  formula  for  —~  completely  determines  the  perturbations 

due  to  the  action  of  the  component  Z  perpendicular  to  the  plane  of  the 
instantaneous  orbit,  instead  of  determining  p'  and  qf  by  an  independent 
integration  by  means  of  the  results  given  by  the  equations  (162),  it 

will  be  preferable  to  derive  them  directly  from  dz,  and  -^-'-  The 
equations  (161)  give 

p'  =  —  cos  &0  da  —  sin  &0  d{3,  q'  =  —  sin  £0  *»  +  cos  &0  dp. 

Substituting  for  £a  and  dfl  their  values  given  by  (125)  and  (126), 
and  putting 

x"  —  x  cos  &,  +  y  sin  £0,  y"  =  —  x  sin  &0  +  y  cos  &0, 

we  obtain 


/    ,,dtz,       .    dx"\ 

r  dt  '  &'W./- 


VAKIATION   OF   CO-OKDINATES.  501 

Substituting  further  the  values 

x"  =  r  cos  (J,  —  &0),  y"  =  r  sin  (J,  — 

and  also 

(W,  _ 

~dt 


-\-m      .        _  &lp  (1  -f-  m)         e  sin  v 

=  -  6  Sin  V  - 


—  -  j 

1  -j-  e  cos  v 
we  easily  find,  since  X,  —  v  =  %, 

,  —  On)    d 


JP  =  —  cos—     0 

ijp« 

,        .  f  .    ,.        ^.x   .       .    /         _  v.fe,  A  —  v      f 

5'  =  +  (Bin  (^-  ft0)  +  esm  (/-  «0))- 


il/p  (1  +  m)      a* 

which  may  be  used  for  the  determination  of  pf  and  qf.  These  equa- 
tions require,  for  their  exact  solution,  that  the  disturbed  values  e,  %, 
and  p  shall  be  known,  but  it  is  evident  that  the  error  will  be  slight, 
especially  when  e  is  small,  if  we  use  the  undisturbed  values  eQ)  p09 
and  £0  =  TTO.  The  actual  values  of  X,  and  r  are  obtained  directly  from 
the  values  of  the  perturbations. 

When  pf  and  qf  have  been  found,  it  remains  only  to  find  cos  i,  and 
1  —  cos  r/,  in  order  to  be  able  to  obtain  F  by  means  of  the  equation 
(159).  From  (153)  we  get 

p'y  -f-  5"  =  sin2  i  —  sin2  i0  —  2qf  sin  i0, 
and  hence 


cos  i  =  1/1—  p'2—  (^  +  sini0)2,  (165) 

from  which  cos*  may  be  found.     The  equation  (157)  gives 

1  —  cos  if  =  cos  iQ  (cos  iQ  +  cos  i)  —  qf  sin  i0,  (166) 

by  means  of  which  the  value  of  1  —  cos  rf  will  be  obtained. 

If  we  substitute  the  values  of  pf,  qf,  -Jp  and  -~  given  by  the 
equations  (153)  and  (162)  in  (159),  it  is  easily  reduced  to 


=  f 
J 


**'  Zdt,  (167; 


(1  —  cos  V)  kVp  (1  +  m) 

which  may  be  used  for  the  determination  of  P.  When  we  neglect 
terms  of  the  order  of  the  cube  of  the  disturbing  force,  in  finding  P 
we  may  use  p0  in  place  of  p  and  put  1  —  cos  rf  =  2  cos2  i0)  so  that  the 
formula  becomes 


502  THEORETICAL   ASTRONOMY. 

Zdt.  (168) 


Cdz, 


187.  By  means  of  the  formulae  which  have  thus  been  derived,  we 
may  find  the  values  of  all  the  quantities  required  in  the  solution  of 
the  equations  (155),  in  order  to  obtain  the  values  of  I  and  b  for  the 
disturbed  motion.  From  r,  I,  and  b  the  corresponding  geocentric 
place  may  be  found.  The  heliocentric  longitude  and  latitude  may 
also  be  determined  directly  by  means  of  the  equations  (145),  provided 
that  &,  0,  and  i  are  known;  and  the  required  formula  for  the  deter- 
mination of  these  elements  may  be  readily  derived.  Thus,  the  equa- 
tions (160)  give,  by  differentiation, 

da,  .  di  dff 

-TT  =  —  sin  ff  cos  i  —^  —  sin  i  cos  a  — rr, 
at  at  at 

d/3  .  di         ,    .  .       dff 

whence 


—      cos  ff  cos  i  -=-  —  sin  i  sm  ff    ,. 
at  at  at 


.    .  dff  da,         .       d0 

sm  i  -=-  =  —  cos  ff  -=T  —  sm  ff  -=-, 
at  at  at 

.  di  da    .  dp 

cosi-j-  =  —  sm  ff  -=-  -j-  cosff-j-. 
dt  dt  dt 

Introducing  the  values  of  ~rr  and  -=7-  already  found  into  these  equa- 
tions, and  putting 


we  obtain 

d**  1  •  •    ^ 

—7-  =  —  —  cot  i  sin  U,  — 

*  (169) 


cos     .  — 


and  also,  since  c?<r  =  cos  i  d& , 

1 sin  (Ay  — 


(170) 


by  means  of  which  the  variations  of  <r,  i,  and  &  due  to  the  action 
of  the  disturbing  forces,  may  be  determined.  The  integral  is  in  each 
case  equal  to  zero  at  the  initial  date  ^  to  which  the  fundamental  os- 
culating elements  belong  and  at  which  the  integration  is  to  com- 
mence. 


VAKIATION  OF   CO-ORDINATES.  503 

If  we  find  i,  and  then  a  —  &  from 

fl  -  •  =  f—  ^7-—-  sin  (^,  -  „)  rZ  «,  (171) 

^  £l/      1       m 


m) 
true  longitude  in  the  orbit  will  be  obtained  from 


T^   .          .  -,  .  « 

It  is  evident  that  since  the  expressions  for  —  T->  -7-,  and  —  ;?  re- 

eft     d£  eft 

quire,  for  an  accurate  solution,  that  the  disturbed  values  i,  0,  and  p 
shall  be  known,  and  require,  besides,  that  three  separate  integrations 
shall  be  performed,  unless  the  perturbations  are  computed  only  in 
reference  to  the  first  power  of  the  disturbing  force,  in  which  case  we 
use  iw  p0,  and  &  0  in  place  of  i,  p,  and  <r,  respectively,  in  the  equations 
(169)  and  (170),  the  action  of  the  component  Z  can  be  considered  in 
the  most  advantageous  manner  by  means  of  the  variation  of  z,  arising 
from  this  component  alone;  and  even  when  only  the  perturbations 
of  the  first  order  are  to  be  determined  it  will  still  be  preferable  to 

derive  dz,  by  the  indirect  process  from  the  expression  for  —  ~,  and  to 

determine  the  heliocentric  place  by  means  of  the  equations  (155). 
When  we  neglect  the  terms  of  the  second  order,  these  equations 
become 

cos  b  cos  (I  —  &0)  =  cos  (A,  —  £0), 

cos  b  sin  (I—  £0)  =  sin  (A,  —  &0)  cos  iQ  —  tan  iQ  —,        (172) 
sin  b  =  sin  (A,  —  £  0)  sin  i0  +  -^-, 

by  means  of  which  I  and  b  are  determined  immediately  from  the  per- 
turbations dM9  v,  and  dz,.  The  peculiar  advantage  of  determining 
the  effect  of  the  action  of  the  component  Z  by  means  of  the  partial 
variation  of  z,  is  apparent  when  we  observe  that  the  expressions  for 

-y-  and  —  7^-  involve  sin  i  as  a  divisor  ;  and  in  the  case  of  orbits  whose 

at  at 

inclination  is  small,  this  divisor  may  be  the  source  of  a  considerable 
amount  of  error. 

188.  The  determination  of  the  perturbations  so  as  to  include  the 
higher  powers  of  the  masses  is  readily  effected  by  moans  of  the  com- 

ddM  d*v  d*3z,      ,  ,          c 

plete  expressions  for  —^-,  ^  and   -^-,  when  the  correct  values  of 

Et  S,  Z}  i,  and  p  are  known.     The  corrected  values  of  i  and  p  — 


504  THEORETICAL   ASTRONOMY. 

which  are  required  only  in  the  case  of  3z,  —  may  be  easily  estimated 
with  sufficient  accuracy,  since  we  require  only  cosi,  while  Vp  ap- 
pears as  the  divisor  of  a  term  whose  numerical  value  is  generally 
insignificant.  To  obtain  the  actual  values  of  E,  S,  and  Z,  the  cor- 
rections to  be  applied  to  N,  Nr,  and  /  must  first  be  determined  by 
means  of  the  formula  (136).  The  values  of  dif  and  dQ>'  will  be 
found  by  means  of  the  data  furnished  by  the  tables  of  the  motion  of 
the  disturbing  body,  and  the  corresponding  corrections  for  Ny  Nft 
and  I  having  been  found  by  "means  of  the  terms  of  (136)  involving 
dif  and  d&',  there  remain  the  corrections  due  to  di  and  S&>  to  be 
applied.  These  may  be  found  in  terms  of  the  quantities  pf  and  q' 
already  introduced.  Thus,  the  equations 

dp'  =  cos  i  sin  (<r  —  ft0)  di  -|-  sin  i  cos  (<r  —  ft  0)  dff, 
dqf  =  cos  i  cos  (<r  —  ft  0)  di  —  sin  i  sin  (<r  —  ft  o)  dff, 
give 

cosi  di  =  sin  (<7  —  ft0)  dp'  -f-  cos  (<r  —  ft0)  d<f, 
sin  id<r  =  cos  (ff  —  ft  0)  dp'  —  sin  (a  —  ft  0)  dqf. 

The  equations  (136)  give,  observing  that  da  —  cosi  dft, 

di    =  —  cos  N  di  —  tan  i  sin  N  da, 

,    ,  sin  N  ..  .        tan  i  ,  • 

dN  =  -j  —  :  —  =-  di  --  :  —  =-  cos  Ndff, 
sin  1  sin  1 

and,  substituting  the  preceding  values  of  di  and  d<r,  these  become 

di    =      sin  (-^  +  ^  —  ^o)  d  ,      cos  (N  +  ff  —  &  o)  ^ 
cos  i  cos  i  ' 


dN,  =  _  CM^-.  ,  , 


sin  I  cos  i  sin  /  cos  i 

If  we  neglect  the  perturbations  of  the  third  order,  these  equations 
give 

cos  i0  cos  i0' 

3N'  =  —  cosec  J(  cos  N-^—r  —  sin  N -±-r  I, 

\  COS  1Q  COS  ?0  / 

by  means  of  which  31  and  3N  may  be  determined,  pr  and  qf  being 
found  by  means  of  the  equations  (164),  using  eot  TTO,  and  p0  in  place 
of  et  ^  and  p.  The  results  for  £7  and  3N'  obtained  from  (173) 
being  applied  to  the  values  of  I'  and  Nf  as  already  corrected  on 
account  of  dif  and  d  3,',  give  the  required  values  of  these  quantities. 


NUMERICAL   EXAMPLE.  505 

When  we  consider  only  di  and  cZ&,  since 

sin  i'  cos  N'  =  cos  i  sin  /  -j-  sin  i  cos  /  cos  N9 

we  easily  find 

dN=  cos  I8N'  —  3*,  (174) 

and  if  we  add  the  quantity  cosIdN'  to  the  value  of  N  already  cor- 
rected on  account  of  Si'  and  S&f,  and  denote  the  result  by  Nn  the 
required  value  of  N  will  be  N,  —  tiff.  -Then,  according  to  (131),  we 
may  compute  w'  +  da  and  /9'  by  means  of  the  formulae 


tan  ((>'  4-  <Jff)  —  N,)  =  tan  <  cos  I,  (175) 

tan  /?'  =  tan  Jsin  ((wr  +  d<r)  —  N,), 

using  the  values  of  N*  and  J  as  finally  corrected.  We  have,  further. 
according  to  (135), 

w  +  dff  =  v,  -f  *0  —  &0> 

by  means  of  which  we  may  compute  the  value  of  w  -}-  d0;  then  the 
value  of  wf  —  w  required  in  the  equations  (132),  and  also  in  finding 
the  value  of  /?,  will  be  given  by 


'  —  w  =  (wr  -j-  da)  —  ( 


w 


and  the  forces  R,  89  and  Z  may  be  accurately  determined. 

By  thus  determining  the  correct  values  of  H,  S,  and  Z  from  date 
to  date,  the  perturbations  3M,  v,  and  dz,  may  be  determined  in  refer- 
ence to  the  higher  powers  of  the  disturbing  forces  according  to  the 
process  already  explained.  The  only  difficulty  to  be  encountered  is 
that  which  arises  from  the  quantities  71,  p'9  and  qf,  required  in  the 
determination  of  the  heliocentric  place  of  the  disturbed  body  by 
means  of  the  equations  (155).  If  an  exact  ephemeris  for  a  short 
period  is  required,  by  means  of  the  complete  perturbations  we  may 
determine  new  osculating  elements,  and  by  means  of  these  the  required 
heliocentric  or  geocentric  places. 

189.  EXAMPLE.  —  We  will  now  illustrate  the  application  of  the 
formulae  for  the  determination  of  the  perturbations  3M,  v,  and  8z,  by 
a  numerical  example;  and  for  this  purpose  let  it  be  required  to 
determine  the  perturbations  of  Eurynome  @  arising  from  the  action 
of  Jupiter  from  1864  Jan.  1.0  to  1865  Jan.  15.0,  Berlin  mean 


506 


THEORETICAL   ASTRONOMY. 


time,  the   fundamental   osculating   elements   being   those   given   in 
Art.  166. 

In  the  first  place,  by  means  of  the  formulae  (130),  using  the  values 


=  206°  39'    5".7, 
'=  98    58  22  .7, 


i=4°  36'52'M, 
i'  =  l    18  40  .5, 


which  refer  to  the  ecliptic  and  mean  equinox  of  1860.0,  we  obtain 
N=  194°  0'  49".9,       N'.=  301°  38'  31".7,       1=  5°  9'  56".4. 

Then,  by  means  of  the  data  furnished  by  the  Tables  of  Jupiter,  we 
find  the  values  of  uf,  the  argument  of  the  latitude  of  Jupiter  in  refer- 
ence to  the  ecliptic  of  1860.0,  and  from  the  equations  (131)  we  derive 
w'  and  f)f.  The  values  of  r'  are  given  by  the  Tables  of  Jupiter,  and 
the  values  of  r0  and  v0  are  found  from  the  elements  given  in  Art. 
166.  The  results  thus  obtained  are  the  following: — 


Berlin  Mean  Time. 

Iogr0 

*V> 

logr1 

ft 

1863  Dec. 

12.0, 

0.294084 

354°  26'  18/x.O 

0.73425 

14°  18'  54".6 

—  0°  l/38".l 

1864  Jan. 

21.0, 

0.294837 

10 

2  45 

.7 

0.73368 

17  21 

44  .2 

0 

18 

9  .1 

March 

1.0, 

0.300674 

25 

24  59 

.4 

0.73305 

20  25 

5  .2 

0 

34 

39  .9 

April 

10.0, 

0.310864 

40 

13  31 

.8 

0.73237 

23  28 

59  .8 

0 

51 

7  .6 

May 

20.0, 

0.324298 

54 

14  41 

.4 

0.73164 

26  33 

32  .1 

1 

7 

29  .7 

June 

29.0, 

0.339745 

67 

21  23 

.5 

0.73086 

29  38 

44  .8 

1 

23 

43  .5 

Aug. 

8.0, 

0.356101 

79 

32  18 

.1 

0.73003 

32  44 

41  .2 

1 

39 

46  .3 

Sept. 

17.0, 

0.372469 

90 

49  57 

.6 

0.72915 

35  51 

24  .6 

1 

55 

35  .2 

Oct. 

27.0, 

0.388214 

101 

19  9 

.8 

0.72823 

38  58 

57  .5 

2 

11 

7  .5 

Dec. 

6.0, 

0.402894 

111 

5  42 

.2 

0.72726 

42  7 

23  .3 

2 

26 

20  .3 

1365  Jan. 

15.0, 

0.416240 

120 

15  32 

.6 

0.72625 

45  16 

43  .9 

2 

41 

10  .6 

The  value  of  w  for  each  date  is  now  found  from 

w  =  v.  +  r0  -  £0  =  v0  +  197°  38'  6".5, 

and  the  cojmponents  of  the  disturbing  force  are  determined  by  means 
of  the  formulae  (132),  p  being  found  from  (133)  or  (134),  and  h  from 
(70).  The  adopted  value  of  the  mass  of  Jupiter  is 


1047.879 

and  the  results  for  the  components  R,  S,  and  Z  are  expressed  in  units 
of  the  seventh  decimal  place.  The  factor  <w2  is  introduced  for  conve- 
nience in  the  integration,  CD  being  the  interval  in  days  between  the 
successive  dates  for  which  the  forces  are  to  be  determined.  Thug  we 
obtain  the  following  results: — 


NUMERICAL   EXAMPLE. 


507 


Date. 

tfR 

u2Sr0 

tJ2Zcost0 

cj\Sr0dt 

1863  Dec. 

12.0, 

4-  70.82 

+     7.16 

-f  0.04 

4-    1.37 

1864  Jan. 

21.0, 

68.95 

—   32.76 

0.49 

-  11.45 

March 

1.0, 

61.16 

70.38 

0.92 

63.32 

April 

10.0, 

48.57 

102.91 

1.32 

150.48 

May 

20.0, 

32.77 

128.34 

1.68 

266.75 

June 

29.0, 

4-  15.41 

145.39 

1.96 

404.35 

Aug. 

8.0, 

-    2.19 

153.44 

2.17 

554.54 

Sept. 

17.0, 

19.12 

152.41 

2.29 

708.21 

Oct. 

27.0, 

34.81 

142.50 

2.25 

856.39 

Dec. 

6.0, 

48.95 

124.04 

2.09 

990.36 

1865  Jan. 

15.0, 

—  61.45 

—    97.36 

-fl.75 

—  1101.73 

The  single  integration  to  find  wlSr0dt  is  effected  by  means  of  tho 
formula  (32). 

The  equations  for  the  determination  of  the  required  differential 
coefficients  are 


ddM 


d*v       <o*R      2a>k*       1         r  e0sinvQ    20      w'P 

-j^:  =  --  --  1  ---  T=  w  I  $.<B  --  -       -<*>S  --  r  v, 
d?         r  r3  J  r* 


, 
=  wZ  cos  l  — 


Substituting  in  these  the-  results  already  obtained,  and  also 

log  fi0  =  2.967809,        log  Po  =  0.371237,        log  e0  =  9.290776, 

we  obtain  first,  by  an  indirect  process,  as  illustrated  in  the  case  of 
the  direct  determination  of  the  perturbations  of  the  rectangular  co- 

ordinates, the  values  of  w2  -^  and  a)2~JpL>  an^  then,  having  found  v, 

w  —rr-  is  given  directly  by  the  first  of  these  equations.    The  integra- 

tion of  the  results  thus  derived,  by  the  formulae  for  mechanical  quad- 
rature, furnishes  the  required  values  of  v,  dM,  and  dz,.  The  calcula- 
tion of  the  indirect  terms  in  the  determination  of  v  and  dz,,  there 
being  but  one  such  term  in  each  case,  is,  on  account  of  the  smallness 
of  the  coefficient,  effected  with  very  great  facility. 
The  final  results  are  the  following  :  — 


508 


THEORETICAL   ASTRONOMY. 


Date. 

it 

dSM 
dt 

&v 

^w 

(W«, 

^-W 

dM 

V 

*, 

1863 

Dec. 

12.0, 

—  0' 

'.028 

+  36.16 

+  0.04 

+  0' 

'.01 

+  4.41 

4-  0.02 

1864 

Jan. 

21.0, 

0 

.072 

33.61 

0.49 

—  0 

.01 

4.31 

0.04 

March 

1.0, 

0 

.499 

22.55 

0.89 

0 

.27 

37.11 

0.54 

April 

10.0, 

1 

.213 

-f    5.58 

1.21 

1 

.11 

91.96 

1.93 

May 

20.0, 

2 

.070 

-  13.52 

1.45 

2 

.75 

152.22 

4.52 

June 

29.0, 

2 

.902 

31.59 

1.53 

5 

.24 

199.05 

8.54 

Aug. 

8.0, 

3 

.546 

46.65 

1.60 

8 

.49 

214.54 

14.10 

Sept. 

17.0, 

3 

.858 

57.88 

1.52 

12 

.22 

183.69 

21.24 

Oct. 

27.0, 

3 

.723 

65.19 

1.28 

16 

.05 

+  95.29 

29.90 

Dec. 

6.0, 

3 

.056 

68.83 

0.92 

19 

.49 

—  58.00 

39.82 

1865 

Jan. 

15.0, 

—  1 

.800 

—  69.19 

+  0.40—21 

.97 

—279.84 

+50.64 

Since,  during  the  period  included  by  these  results,  the  perturbations 
of  the  second  order  are  insensible,  we  have,  for  the  perturbations  of 
Eurynome  arising  from  the  action  of  Jupiter  from  1864  Jan.  1.0  to 
1865  Jan.  15.0, 


3  M  =  —  21".97, 


=  —  0.00002798, 


,  =  +  0.00000506. 


It  is  to  be  observed  that  dz,  is  not  the  complete  variation  of  the  co- 
ordinate zf  perpendicular  to  the  ecliptic,  but  only  that  part  of  this 
variation  which  is  due  to  the  action  of  the  component  Z  alone;  and 
hence  the  results  for  dz,  differ  from  the  complete  values  obtained 
when  we  compute  directly  the  variations  of  the  rectangular  co- 
ordinates. 

Let  us  now  determine  the  heliocentric  longitude  and  latitude  for 
1865  Jan.  15.0,  Berlin  mean  time,  including  the  perturbations  thus 
derived.  From  the  equations 


E,  —  e0  sin  E,  =  M,, 
r,    =a0(l  —  e0cosJ£,), 

sin  J  (v,  —  E,)  =  sin  £  ?0  sin  E, 


we  obtain 


M,      =  99°  29'  35".51, 
log  r,  =  0.4162304, 
log  r  =  0.4162183, 


£,=  110°  0'33".75, 
v,  =  120  15  13  .80, 
J  =  164  32  25  .97. 


The  calculation  of  the  values  of  r,  and  v,  from  the  values  of  Mf9  a0, 
and  e0,  may  be  effected  by  means  of  the  various  formulae  for  the 


NUMERICAL   EXAMPLE.  509 

determination  of  the  radius-vector  and  true  anomaly  from  given 
elements.  If  we  substitute  these  results  for  ^,,  r,  and  dz,  in  the  equa- 
tions (172),  we  get 

I  =  164°  37'  59".05,  b  =  —  3°  5'  32".54, 

which  are  referred  to  the  ecliptic  and  mean  equinox  of  1860.0,  and 
from  these  we  may  derive  the  geocentric  place  of  the  disturbed  body. 
If  the  place  of  the  body  is  required  in  reference  to  the  equinox  and 
ecliptic  of  any  other  date,  it  is  only  necessary  to  reduce  the  elements 
x0,  QQ,  and  i0  to  the  equinox  and  ecliptic  of  that  date;  and  then, 
having  computed  X,  and  r,  we  obtain  by  means  of  the  equations  (172) 
the  required  values  of  I  and  6.  In  the  determination  of  the  pertur- 
bations it  will  be  convenient  to  adopt  a  fixed  equinox  and  ecliptic 
throughout  the  calculation ;  and  afterwards,  when  the  heliocentric  or 
geocentric  places  are  determined,  the  proper  corrections  for  precession 
and  nutation  may  be  applied. 

In  order  to  compare  the  results  obtained  from  the  perturbations 
8M,  v,  and  dz,  with  those  derived  by  the  method  of  the  variation  of 
rectangular  co-ordinates,  we  have,  for  the  date  1865  Jan.  15.0, 

x0  =  —  2.5107584,        y0  =  +  0.6897713,        z0  =  —  0.1406590 ; 
and  for  the  perturbations  of  these  co-ordinates  we  have  found 

8x  =  +  0.0001773,         3y  =  +  0.0001992,        dz  =  —  0.0000028. 
Hence  we  derive 

x  =  —  2.5105811,         y=  +  0.6899705,         z  =  —  0.1406618, 
and  from  these  the  corresponding  polar  co-ordinates,  namely, 
log  r  =  0.4162182,         I  =  164°  37'  59".05,         b  =  —  3°  5'  32".54, 

from  which  it  appears  that  the  agreement  of  the  results  obtained  by 
the  two  methods  is  complete. 

190.  When  the  perturbations  become  so  large  that  the  terms  of  the 
second  order  must  be  retained,  the  approximate  values  which  may  be 
obtained  for  several  intervals  in  advance  by  extending  the  columns 
of  differences,  will  serve  to  enable  us  to  consider  the  neglected  terms 
partially  or  even  completely,  and  thus  derive  the  complete  perturba- 
tions for  a  very  long  period.  But  on  account  of  the  increasing  diffi- 
culties which  present  themselves,  arising  both  from  the  consideration 


510  THEORETICAL   ASTRONOMY. 

of  the  perturbations  due  to  the  action  of  the  component  Z  in  com- 
puting the  place  of  the  body,  and  from  the  magnitude  of  the  numeri- 
cal values  of  the  perturbations,  it  will  be  advantageous  to  determine, 
from  time  to  time,  new  osculating  elements  corresponding  to  the 
values  of  the  perturbations  for  any  particular  epoch,  and  thus  com- 
mencing the  integrals  again  with  the  value  zero,  only  the  terms  of 
the  first  order  will  at  first  be  considered,  and  the  indirect  part  of  the 
calculation  will,  on  account  of  the  smallness  of  the  terms,  be  effected 
with  great  facility.  The  mode  of  effecting  the  calculation  when  the 
higher  powers  of  the  masses  are  taken  into  account  has  already  been 
explained,  and  it  will  present  no  difficulty  beyond  that  which  is  in- 
separably connected  with  the  problem.  The  determination  of  F9  pf, 

and  qr  may  be  effected  from  the  results  for  -77-,  -r-,  and  -~r  by  means 

dt    dt  dt     J 

of  the  formulae  for  integration  by  mechanical  quadrature,  as  already 
illustrated,  or  we  may  find  F  by  a  direct  integration,  and  the  values 

of  pf  and  qf  by  means  of  the  equations  (164),  —^-  being  found  from 

-^-  by  a  single  integration.     The  other  quantities  required  for.  the 

complete  solution  of  the  equations  for  the  perturbations  will  be 
obtained  according  to  the  directions  which  have  been  given;  and  in 
the  numerical  application  of  the  formulae,  particular  attention  should 
be  given  to  the  homogeneity  of  the  several  terms,  especially  since,  for 
convenience,  we  express  some  of  the  quantities  in  units  of  the  seventh 
decimal  place,  and  others  in  seconds  of  arc. 

The  magnitude  of  the  perturbations  will  at  length  be  such  that, 
however  completely  the  terms  due  to  the  squares  and  higher  powers 
of  the  disturbing  forces  may  be  considered,  the  requirements  of  the 
numerical  process  will  render  it  necessary  to  determine  new  osculating 
elements;  and  we  therefore  proceed  to  develop  the  formulae  for  this 
purpose. 


191.  The  single  integration  of  the  values  of  a)2  -p-  and  co2  —^-  will 
give  the  values  of  o>  -57-  and  CD  —jr*  and  hence  those  of  -j  and  -rr> 
which,  in  connection  with  —  ^—  ,  are  required  in  the  determination  of 

the  new  system  of  osculating  elements.     Since  r2-^  represents  double 
the  areal  velocity  in  the  disturbed  orbit,  we  have 


CHANGE  OF   THE   OSCULATING   ELEMENTS.  511 


dv,  _  TcVp(l-\-m) 
dt  ~       ~7» 


The  equation  (109)  gives 


dv,  _kVPo(l+m)  I          1     d9M\ 
'dt  ~  r?  \     %/    dt    r 

Hence,  since  r  =  r,  (1  +  v),  we  obtain 


by  means  of  which  we  may  derive  p.     This  formula  will  furnish  at 
once  the  value  of  p,  which  appears  in  the  complete  equation  foi 

—TOT*  and  also  in  the  equations  (164);  and  the  value  of  cost  may  be 

determined  by  means  of  (165). 
In  the  disturbed  orbit  we  have 

dr       kyU  4-  m 


and  the  equations  (108)  and  (111)  give 
dr 


Therefore  we  obtain 

dv 


which,  by  means  of  (176),  becomes 


The  relation  between  r  and  r,  gives 


P  __  __.         Po         /     I 


1  -f  e  cos  v      1  +  e0  cos  v, 
and,  substituting  in  this  the  value  of  p  already  found,  we  get 

«,  cos.,,)  (  1  +  1  .  ^  J  (1  +  v)'  -  1.         (178) 


e  cos  v  — 


512  THEORETICAL   ASTRONOMY. 

Let  us  now  put 


(179) 


a  and  /5  being  small  quantities  of  the  order  of  the  disturbing  force, 
and  the  equations  (177)  and  (178)  become 

e  sin  v  =  e0  sin  vt  -\-  ae0  sin  v,  -f-  /?, 
e  cos  v  =  e0  cos  v,  -f-  ae0  cos  v,  -f-  a. 

These  equations  give,  observing  that  r,  (cos  v,  -f-  e0)  =p0  cos  .#„ 

e  sin  (v,  —  v)  =  a  sin  v,  —  ft  cos  vn 

e  cos  (v,  —  v)  =  e0  -f  —  cos  E,-\-  P  sin  v,, 
^*/ 

from  which  e,  v,  —  v,  and  ?;  may  be  found;  and  thus,  since 

*  =  *  +  («,—«),  (181) 

we  obtain  the  values  of  the  only  remaining  unknown  quantities  in 
the  second  members  of  the  equations  (164).  The  determination  of 
pf  and  (f  may  now  be  rigorously  effected,  and  the  corresponding 

value  of  cosi  being  found  from  (165),  -£-  and  -g-  will  be  given  by 

(162).  Then,  having  found  also  1  —  cos  37'  by  means  of  (166),  F  may 
be  determined  rigorously  by  the  equation  (159),  and  not  only  the 
complete  values  of  the  perturbations  in  reference  to  all  powers  of  the 
masses,  hut  also  the  corresponding  heliocentric  or  geocentric  places 
of  the  body,  may  be  found. 
If  we  put 

Y*  =  a  sin  v,  —  p  cos  vn 


and  neglect  terms  of  the  third  order,  the  equations  (180)  give 


r'         * 

v  —v  =  —  s  —  —  s, 
e,          ej 

in  which  s  =  206264".8.     These  equations  are  convenient  for  the 


CHANGE   OF   THE   OSCULATING  ELEMENTS.  513 

determination  of  e  and  v,  —  v,  and  hence  7.  by  means  of  (181),  when 
the  neglected  terms  are  insensible. 

The  values  of  p,  e,  and  v  having  been  found,  we  have 


m 


, 

a2 

tan  A  E  =  tan  (45°  —  £  ?)  tan  %  v,  M=  E—esin  E, 

from  which  to  find  the  elements  <pt  a,  //,  and  M.  The  mean  anomaly 
thus  found  belongs  to  the  date  t,  and  it  may  be  reduced  to  any  other 
epoch  denoted  by  t0  by  adding  to  it  the  quantity  p.  (tQ  —  t).  When  we 
neglect  the  terms  of  the  third  order,  we  have 

«_<,  =  _  sin  y-  sin  y, 

cos?n  —  4(?  —  Po)sin?0 

and  if  we  substitute  for  sin  tp  —  sin  <pQ  =  e  —  e0  the  value  given  bv 
the  first  of  equations  (183),  the  result  is 

2*  Bin  y0 


2  sin  <pQ  cos  9?0  —  df  sin  <pti  tan 
from  which  we  get 

dr  3'*  sin  <pn  Y'Z 


s- 

by  means  of  which  y>  may  be  found  directly,  terms  of  the  third  order 
being  neglected. 

In  the  case  of  the  orbits  of  comets  for  which  e  differs  but  little 
from  unity,  instead  of  dM  we  compute  by  means  of  the  formula 
(142)  the  value  of  dT,  and  since  we  have 

ddT  _    _!_   dSM 
dt  n0      dt 

the  equation  for  p  becomes 

(jxm  \2 
l-^f)  (!  +  ")';  (186) 

and  for  a  we  have 


Then  e,  t/,  and  q  will  be  found  by  means  of  the  equations 


514  THEORETICAL   ASTRONOMY. 

e  sin  (Vf  —  v)  =  a  sin  v,  —  /5  cos  v,, 

e  cos  (v,  —  v)  =  eQ  -f  a  (cos  v,  +  e0)  +  /5  sin  v,,  (188) 

P 

«=IT7 

and  the  time  of  perihelion  passage  will  be  derived  from  e  and  v  by 
means  of  Table  IX.  or  Table  X. 

There  remain  yet  to  be  found  the  elements  <r,  &,  and  i,  which  de- 
termine the  position  of  the  plane  of  the  disturbed  orbit  in  space. 
The  values  of  pf  and  q'  will  be  found  from  the  equations  (164),  and 
P,  whenever  it  may  be  required,  will  be  determined  as  already 
explained.  Then  we  shall  have 

sin  i  sin  (<r  —  &  0)  =  pr  ,  (189) 

sin  i  cos  (a  —  &0)  =  q'  -f  sin  iQ, 

from  which  to  find  i  and  a.    When  we  neglect  the  terms  of  the  third 
order,  these  equations  give 

sin  i  —  sin  in  =  q'  4-    .     .  » 
a        smt0 

and  hence 


sm  10 


t  ,, 

r  cos  IQ      '   2  cos3  IQ      '   2  sm  t0  cos  IQ 

in  which  s  =  206264".  8.     The  auxiliary  spherical  triangle  which  we 
have  employed  in  the  derivation  of  the  equations  (155)  gives  directly 

cos  %(i  -f-  i0)  _  tan|(<r—  &0) 


cos  £  (i  —  *0)       tan  ^  (&  —  h  -f  h0  —  &  0) 
and  since  h  —  h0  =  F,  we  have 

tanK^-^o-O  =  ^|^~^tanj(^-^o)>       (191) 

by  means  of  which  the  value  of  &  may  be  found.     This  equation 
gives,  when  we  neglect  terms  of  the  third  order, 

«  =  «'+r+^°  +  2^-^-«.).      ^2) 

Substituting  in  this  the  values  of  ff—  &0  and  i  —  i0  given  by  (190), 
we  get 

1~3°iri2/1  (193) 

sin  i0  cos  i,         smz  IQ  -~~a  -  "  * 


CHANGE   OF  THE   OSCULATING   ELEMENTS.  515 

r  being  expressed  in  seconds  of  arc.  Finally,  for  the  longitude  of; 
the  perihelion,  we  have 

*=*+£—*,  (194)' 

and  the  elements  of  the  instantaneous  orbit  are  completely  deter- 
mined. When  we  neglect  terms  of  the  third  order,  this  equation, 
substituting  the  values  given  by  (190)  and  (192),  becomes 


It  should  also  be  observed  that  the  inclination  i  which  appears  in 
these  formulae  is  supposed  to  be  susceptible  of  any  value  from  0°  to 
180°,  and  hence  when  i  exceeds  90°  and  the  elements  are  given  in 
accordance  with  the  distinction  of  retrograde  motion,  they  are  to  be 
changed  to  the  general  form  by  using  180°  —  i  instead  of  i,  and; 
2&  —  TT  instead  of  TT. 

The  accuracy  of  the  numerical  process  may  be  checked  by  com- 
puting the  heliocentric  place  of  the  body  for  the  date  to  which  the 
new  elements  belong  by  means  of  these  elements,  and  comparing  the 
results  with  those  obtained  directly  by  means  of  the  equations  (155). 
We  may  remark,  also,  that  when  the  inclination  does  not  differ  much 
from  90°,  the  reduction  of  the  longitudes  to  the  fundamental  plane 
becomes  uncertain,  and  F  may  be  very  large,  and  hence,  instead  of 
the  ecliptic,  the  equator  must  be  taken  as  the  fundamental  plane  to 
which  the  elements  and  the  longitudes  are  referred. 

192.  Although,  by  means  of  the  formulae  which  have  been  given, 
the  complete  perturbations  may  be  determined  for  a  very  long  period 
of  time,  using  constantly  the  same  osculating  elements,  yet,  on 
account  of  the  ease  with  which  new  elements  may  be  found  from  dM, 


.  ,  f    .      f    ..         „.    ,   ,   . 

v,  oz,,  -ji-»  -ji»  and  —fj->  and  on  account  of  the  facility  afforded  in. 

the  calculation  of  the  indirect  terms  in  the  equations  for  the  differen- 
tial coefficients  so  long  as  the  values  of  the  perturbations  are  small, 
it  is  evident  that  the  most  advantageous  process  will  be  to  compute 
8M,  v,  and  8z,  only  with  respect  to  the  first  power  of  the  disturbing 
force,  and  determine  new  osculating  elements  whenever  the  terms  of' 
the  second  order  must  be  considered.  Then  the  integration  will 
Hgain  commence  with  zero,  and  will  be  continued  until,  on  account 
of  the  terms  of  the  second  order,  another  change  of  the  elements  is 
required.  The  frequency  of  this  transformation  will  necessarily  de- 


5io  THEORETICAL   ASTRONOMY. 

pend  on  the  magnitude  of  the  disturbing  force;  and  if  the  disturbed 
body  is  so  near  the  disturbing  body  that  a  very  frequent  change  of 
the  elements  becomes  necessary,  it  may  be  more  convenient  either  to 
include  the  terms  of  the  second  order  directly  in  the  computation 
of  the  values  of  dM,  vy  and  dzn  or  to  adopt  one  of  the  other  methods 
which  have  been  given  for  the  determination  of  the  perturbations  of 
a  heavenly  body.  In  the  case  of  the  asteroid  planets,  the  consider- 
ation of  the  terms  of  the  second  order  in  this  manner  will  only 
require  a  change  of  the  osculating  elements  after  an  interval  of  seve- 
ral years,  and  whenever  this  transformation  shall  be  required,  the 
equations  for  <p,  i,  &,  and  TT,  in  which  the  terms  of  the  third  order 
are  neglected,  may  be  employed.  It  should  be  observed,  however, 
that  the  perturbations  of  some  of  the  elements  are  much  greater  than 
the  perturbations  of  the  co-ordinates,  and  hence  when  terms  depend- 
ing on  the  squares  and  higher  powers  of  the  masses  have  been 
neglected  in  the  computation  of  these  perturbations,  it  may  still  be 
necessary  to  include  the  values  of  the  terms  of  the  second  order  in 
the  incomplete  equations  referred  to.  No  general  criterion  can  be 
given  as  to  the  time  at  which  a  change  of  the  osculating  elements 
will  be  required;  but  when,  on  account  of  the  magnitude  of  the 
values  of  dM,  v,  and  3z,9  it  appears  probable  that  the  perturbations 
of  the  second  order  ought  to  be  included  in  the  results,  by  computing 
a  single  place,  taking  into  account  the  neglected  terms,  we  may  at 
once  determine  whether  such  is  the  case  and  whether  new  elements 
are  required. 

193.  We  have  already  found  the  expressions  for  the  variations  of 
££  and  i  due  to  the  action  of  the  disturbing  forces,  and  we  shall  now 
consider  those  for  the  variation  of  the  other  elements  of  the  orbit 
directly.  Let  x,  T/,  z  be  the  co-ordinates  of  the  body  at  any  given 
time  referred  to  any  fixed  system  of  co-ordinates.  These  will  be 
known  functions  of  the  six  elements  of  the  orbit  and  of  the  time. 
Tf  the  body  were  not  subject  to  the  action  of  the  disturbing  forces, 
these  six  elements  would  be  rigorously  constant,  and  the  co-ordinates 
would  vary  only  with  the  time ;  but  on  account  of  the  action  of  these 
forces  the  elements  must  be  regarded  as  continuously  varying  in  order 
that  the  relation  between  the  elements  and  the  co-ordinates  at  any 
instant  shall  be  expressed  by  equations  of  the  same  form  as  in  the 
case  of  the  undisturbed  motion.  The  co-ordinates  will,  therefore,  in 
the  disturbed  motion,  be  subject  to  two  distinct  variations:  that 
which  results  from  considering  the  time  alone  to  vary,  and  that  which 


VARIATION   OF   CONSTANTS.  517 

results  from  the  variation  of  the  elements  themselves.    Let  these  two 

kinds  of  partial  variations  be  symbolized  respectively  by  I  -5-  )  and 

[j  ~\  \  ®   i 

•jg-j*  and  similarly  in  the  case  of  the  other  co-ordinates;  then  will 

the  total  variations  be  given  by 


dx_ 
~dt'~\~dt 


—    —  _ 

dt         dt~~dt 


But  if  we  differentiate  twice  in  succession  the  equations  which  ex- 
press the  values  of  x,  y,  and  z  as  functions  of  the  elements  and  of 
the  time,  regarding  both  the  elements  and  the  time  as  variable,  the 
substitution  of  the  results  in  the  general  equations  for  the  motion  of 
the  disturbed  body  will  furnish  three  equations  for  the  determination 
of  the  variations  of  the  elements.  There  are,  however,  six  unknown 
quantities  to  be  determined;  and  hence  we  may  assign  arbitrarily 
three  other  equations  of  condition.  The  supposition  which  affords 
the  required  facility  in  the  solution  of  the  problem  is  that 


and  hence  that 

<te__(d*_\  ^L  —  [^_\  $L  —  [<*L\ 

dt       \  dt  I  dt       \  dt  r  dt  ~\  dt  /' 

It  thus  appears  that  in  order  that  the  integrals  of  the  equations  (1) 
shall  be  of  the  same  form  as  those  of  the  equations  (3),  —  the  arbi- 
trary constants  of  integration  which  result  from  the  integration  of 
the  latter  being  regarded  as  variable  when  the  disturbing  forces  are 
considered,  —  the  first  differential  coefficients  of  the  co-ordinates  with 
respect  to  the  *ime  have  the  same  form  in  the  disturbed  and  undis- 

turbed orbits.     But  since  ^jp  H|I  and  -g-  are  the  velocities  of  the 

disturbed  body  in  directions  parallel  to  the  co-ordinate  axes  respect- 
ively, it  follows  that  during  the  element  of  time  dt  the  velocity  of 
the  body  must  be  regarded  as  constant,  and  as  receiving  an  increment 
only  at  the  end  of  this  instant.  The  equations  (197)  show  also  that 
\f  we  differentiate  any  co-ordinate,  rectangular  or  polar,  referred  to  a 


518  THEORETICAL   ASTRONOMY. 

fixed  plane  and  measured  from  a  fixed  origin,  with  respect  to  the  ele~ 
ments  alone  considered  as  variable,  the  first  differential  coefficient 
must  be  put  equal  to  zero,  and  this  enables  us  at  once  to  effect  the 
solution  of  the  problem  under  consideration.  It  is  to  be  observed, 
further,  that  the  functions  whose  first  differential  coefficients  with 
respect  to  the  time  when  only  the  elements  are  regarded  as  variable 
are  thus  put  equal  to  zero,  must  not  involve  directly  the  motion  of 
the  disturbed  body,  since  the  second  differential  coefficients  of  the  co- 
ordinates have  not  the  same  form  in  the  case  of  the  disturbed  motion 
as  in  that  of  the  undisturbed  motion. 

194.  If  we  suppose  the  disturbing  force  to  be  resolved  into  three 
components,  namely,  It  in  the  direction  of  the  disturbed  radius- 
yector,  S  in  a  direction  perpendicular  to  the  radius-vector  and  in  the 
plane  of  disturbed  orbit,  positive  in  the  direction  of  the  motion,  and 
fi  perpendicular  to  the  plane  of  the  instantaneous  orbit,  the  latter 
will  only  vary  &  and  i  and  the  longitude  of  the  perihelion  so  far  as 
jt  is  affected  by  the  change  of  the  place  of  the  node,  while  the  forces 
JR  and  S  will  cause  the  elements  M,  TT,  6,  and  a  to  vary  without  affect- 
ing Q>  and  ?'. 

Let  us  now  differentiate  the  equation 


regarding  the  elements  as  variable,  and  we  get 

2rdrl_        _1_     da  2V          l 

r»  L  dt  J  "          a2  '   dt  +  k2  (1  +  m)  '    dt 

T 


dt       tf  (1  +  m)      dt 

The  differential  coefficient  —  is  here  the  increment  of  the  accele- 

dt 

rating  force,  in  the  direction  of  the  tangent  to  the  orbit  at  the  given 
point, due  to  the  action  of  the  disturbing  force;  and  if  we  designate 
the  angle  which  the  tangent  makes  with  the  prolongation  of  the 
radius-vector  by  ^0,  we  shall  have 

_—  =  E  cos  4'0  +  S  sin  4'v 
dt 

-Substituting  this  value  in  the  preceding  equation,  we  obtain 


VARIATION   OF   CONSTANTS.  519 


But  we  have,  according  to  the  equations  (50)6, 


dv 


in  wnich  v  denotes  the  true  anomaly  in  the  instantaneous  orbit;  and 
hence  there  results 


e  sn 


by  means  of  which  the  variation  of  a  may  be  found. 
If  we  introduce  the  mean  daily  motion  ft,  we  shall  have 

•^r  =  —  ^'-^p  (199) 

and  hence 

?  sin  vR  +  2-  £),  (200) 


for  the  determination  of  dp. 

The  first  of  the  equations  (97)  gives 


and  hence  we  obtain 

d  (i/p)  = 

<ft 
or  _ 

(201 


m 
The  equation  p  =  a  (1  —  e2)  gives 

*    _.    -P          <*«          Oflfi—  . 

"3T"-  5T    "it          6  ^ 

Equating  these   values   of  -^-»   and  introducing  the  value  of  -^ 
already  found,  we  get 


-_.  (202) 

o 


520  THEORETICAL   ASTRONOMY. 

and  since 

—  =  1  -{-  e  cos  v,  —  =  1  —  e  cos  E, 

T  CL 

E  being  the  eccentric  anomaly  in  the  instantaneous  orbit,  this  becomes 
de  1 


(p  sin  vR  -f  p  (cos  v  +  cos  E)  S),       (203) 


which  will  give  the  variation  of  e.     If  we  introduce  the  angle  of 
eccentricity  <p,  we  shall  have 


and  hence 
d<p 


de  dy 

—  =  cos<p  -^,  p  =  a  cos2  ?t 


7=  (a  cos  <p  sin  vE  -f-  «  cos  ^  (cos  v  -f-  cos  J£)  #).     (204) 


195.  When  we  consider  only  the  components  R  and  8  of  the  dis- 
turbing force,  the  longitude  in  the  orbit  will  be 


We  have,  therefore, 

.£-=1 

the  differentiation  of  which,  regarding  the  elements  as  variable,  gives 
dp       pT  dr~\  ..         .  de 

-- 


or 

dp  de  .       d% 

_JL  =  rcosVw  +  ersmv_ 

Therefore 


and,  since  p  cos  ^=  r  (cos  v  +  e),  we  have 

p  (1  —  cos  v  cos  E)=r  sin*  v, 
so  that  the  equation  becomes 


VARIATION   OF   CONSTANTS.  521 

^(-pco*VIi  +  (p+r)s[nvS)>  (205) 

from  which  the  value  of  ~-  may  be  derived. 

at 

If  we  introduce  the  element  co,  or  the  angular  distance  of  the  peri- 
helion from  the  ascending  node,  it  will  be  necessary  to  consider  also 
the  component  Z;  and,  since  co  =  X,  —  <r,  we  shall  have 

dot  __  d%        dff        d%  .  d& 

~dt==-dt^-dt==-dt-  C08*-ST> 


and  hence 


(206) 


In  the  case  of  the  longitude  of  the  perihelion,  we  have 


dt         dt          dt 
and  therefore 

dn  1  1 

•  —  ( — p  cos  vR  +  (p 


sin2  Ji^£.  (207) 

The  first  of  the  equations  (15)2  gives 

dt*,  de 


in  which  MQ  denotes  the  mean  anomaly  at  the  epoch,  which  is  usually 
adopted  as  one  of  the  elements  in  the  case  of  an  elliptic  orbit.  Sub- 
stituting for  -=r  and  -=-  the  values  already  found,  we  get 


dt  dt 

dM 


Lo 


{ (p  cot  <p  cos  v  —  2r  cos  y>)  It 


f—  (2  —  cos2  v  —  cos  v  cos  E}  cot  y  S]  —  (t  —  O-> /> 

sin  v  ^  dt 

or 

dM  1 

-     °.  = —  (( p  cot  <p  cos  v  —  2r  cos  ^)  R  —  (p  -\-  7)  cot  <p  sin  v/S) 

w         kvp  (1  +  ?n-) 

—  (*  —  g  -^-  (208) 

Thp  equation  (205)  gives 


522  THEOKETICAL   ASTRONOMY. 


cot  <p  sinvS  — —  p  cot  <f>  cos  vR 


d/ 

-f  cos  <p  -£-, 
dt 

by  means  of  which  (208)  reduces  to 

""*=«-«_«,)*         (209) 


which  will  determine  the  variation  of  the  mean  anomaly  at  the 
epoch. 

Since  the  equations  for  the  determination  of  the  place  of  the  body- 
in  the  case  of  the  disturbed  motion  are  of  the  same  form  as  those  for  the 
undisturbed  motion,  the  mean  anomaly  at  the  time  t  will  be  given  by 

M=  M0  +  dM0  +  (t  -  «  00  +  ^), 

in  which  //0  denotes  the  mean  daily  motion  at  the  instant  t0.     There- 
fore we  shall  have 


M  =  M0  +  dt  +  va(t-  <„)  +  (t  -  t,)          dt, 

the  integrals  being  taken  between  the  limits  tQ  and  t.     The  quantity 


expresses  the  mean  anomaly  at  the  time  t  in  the  undisturbed  orbit  ; 
and  if  we  designate  by  dM  the  correction  to  be  applied  to  this  IP 
order  to  obtain  the  mean  anomaly  in  the  disturbed  orbit,  so  that 


we  shall  have 
and  hence 


Differentiating  this  with  respect  to  f,  we  get 


VARIATION   OF   CONSTANTS.  523 

Substituting  in  this  the  value  of  —~  from  (209),  the  result  is 
dM  _  __  dx  2r  cos  ?  rd/JL 


=  E  +  f^  eft,          (210) 

l  */    at 


which  does  not  involve  the  factor  t  —  10  explicitly,  and  by  means  of 
which  the  mean  anomaly  in  the  disturbed  orbit,  at  any  instant  t,  may 
be  found  directly  from  that  for  the  same  instant  in  the  undisturbed 
orbit. 

To  find  the  variation  of  the  mean  longitude  L,  we  have 


dL      dM       d*       dx.dM  dQ, 


and  therefore 
dL 


o  •  *  ,  -  „   .  .   ,,        ,_.,, 

2  sm'  -*      '  ~  +  (    } 


To  find  the  variations  of  &  and  i,  since 

u  =  I,  —  <r, 

u  denoting  the  argument  of  the  latitude  in  the  disturbed  orbit,  we 
have,  according  to  the  equations  (169)  and  (170), 


r  sin  u  „ 

m)  '    sin  i      ' 

(212) 

-  T  COS  UZ. 


The  inclination  i  may  have  any  value  from  0°  to  180°;  and  when- 
ever the  elements  are  given  in  accordance  with  the  distinction  of  re- 
trograde motion,  they  must  be  converted  into  those  of  the  general 
form  by  taking  180°  —  i  in  place  of  the  given  value  of*,  and  2&  —  it 
in  place  of  the  given  value  of  TT,  before  applying  the  formula  which 
involve  these  elements. 

196.  In  the  case  of  the  orbits  of  comets  in  which  the  eccentricity 
differs  but  little  from  that  of  the  parabola,  the  perturbations  of  the 
perihelion  distance  q  and  of  the  time  of  perihelion  passage  T  will  be 
determined  instead  of  those  of  the  elements  M  and  a  or  //. 

The  equation 

P  =  9  C1  +  <0 
gives 


524  THEOKETICAL   ASTRONOMY. 

dq  _        1          dp^  q          de 

~dt  ~~  1  +  e  '  ~dt  ~~  1  +  e  '  "3P 

and  substituting  in  this  the  value  of  -rr  already  found,  and  neglect- 
ing the  mass  of  the  comet,  which  is  always  inconsiderable,  we  get 

<fy         V  q         de 

~==  ~"' 


by  means  of  which  the  variation  of  q  may  be  found.  In  the  case  of 
elliptic  motion  the  value  of  -jr  may  be  found  by  means  of  (202)  or 

(203);  but  in  the  case  of  hyperbolic  motion  the  equation  (202)  will 
be  employed.  It  should  be  observed,  also,  that  when  the  general 
formula?  for  the  ellipse  are  applied  to  the  hyperbola,  the  semi- 
transverse  axis  a  must  be  considered  negative. 

When  the  orbit  is  a  parabola,  the  equation  (202)  becomes 

fjf  1 

~  =  -4-  dp  sin  vR  +  2p  cos'  ivtf  ),  (214) 

and  for  the  value  of  -jr  we  have 
at 


It  remains  now  to  find  the  formula  for  the  variation  of  the  time  of 
perihelion  passage.     The  relation  between  T  and  M0  is  expressed  by 

360°  —  J£  =  /i(r—  «, 

the  differentiation  of  which  gives 

dMQ  dp.         dT. 

' 


and,  substituting  for  —~  the  value  given  by  equation  (209),  we  ge* 

dT_2ar          aVp      dx  1      d^ 

W  =  =  "FJ       "F"  "  dt  H          *'*"&' 

Substituting  further  the  values  of  -37  and  -^7  given  by  the  equations 
(205)  and  (199),  the  result  is 


VARIATION   OF   CONSTANTS.  525 

dT       aR  p  Sk(t—  T) 

'dt==~¥^r~eC°SV~~    I/"        esmv) 

VP  (216) 

.    aS  Ip  +  r  .  Zk(t—T)p\ 

H-  ~JT  I       —  sm  v  --     /—       •  —  I' 
V  \     e  i/p          r  I 

which  may  be  employed  to  determine  the  variation  of  T  whenever 
the  eccentricity  is  not  very  nearly  equal  to  unity.  It  is  obvious, 
however,  that  when  a  is  very  large  this  equation  will  not  be  con- 
venient for  numerical  calculation,  and  hence  a  further  transformation 

of  it  is  desirable.  Thus,  if  we  derive  the  expressions  for  -r-  and  -j- 
from  the  equations  (24)2  and  (23)2,  we  easily  obtain 


2p         dr  p  3&(£—  T)  p* 

•r-f-  -  -j-  =  a(2r  —  —  cosv  --  *-?-  —  -  e  sm  v)  +  —^-.  —  75  cos  v, 

1  -f-  e      de  e  y  e  (1  -\-  e)z 


2p        dv          Ip  +  r  .  3k(t  —  T)     p\  p2       I         r\  . 

—1  —  r  -=-  =  a  I  —  —  sm  v  --  ^    -       •  -  )  -  —   1  +  -    smv. 

1  +  e      de          \     e  ]/p  rj       e(l  +  e)2\     ^pl 

By  means  of  these  results  the  equation  (216)  is  transformed  into 


which  may  be  used  for  the  determination  of  -TT-»  the  values  of  -j~ 

ut  de 

and  -r-  being  found  by  means  of  the  various  formula?  developed  in 

Art.  50.  When  a  is  very  large,  its  reciprocal  denoted  by  /may  often 
be  conveniently  introduced  as  one  of  the  elements,  and,  for  the  deter- 
mination of  the  variation  of/,  we  derive  from  equation  (198) 


(218) 


In  the  case  of  parabolic  motion  we  have  6  =  1,  and  p  =  2q;  and 
if  we  substitute  in  (217)  for  -^  and  -^  the  values  given  by  the  equa- 
tions (33)2  and  (30)2,  the  result  is 


(219) 


526  THEORETICAL    ASTRONOMY. 

197.  Instead  of  the  elements  usually  employed,  it  may  be  desirable, 
in  rare  and  special  cases,  to  introduce  other  combinations  of  the  ele- 
ments or  constants  which  determine  the  circumstances  of  the  undis- 
turbed motion,  and  the  relation  between  the  new  elements  adopted 
and  those  for  which  the  expressions  for  the  differential  coefficients 
have  been  given,  will  furnish  immediately  the  necessary  formulae. 
In  the  case  of  the  periodic  comets,  it  will  often  be  desired  to  deter- 
mine the  alteration  of  the  periodic  time  arising  from  the  action  of  the 
disturbing  planets.  Let  us,  therefore,  suppose  that  a  comet  has  been 
identified  at  two  successive  returns  to  the  perihelion,  and  let  T  denote 
the  elapsed  interval.  The  observations  at  each  appearance  of  the 
comet,  however  extended  they  may  be,  will  not  indicate  with  certainty 
the  semi-transverse  axis  of  the  orbit,  and  hence  the  periodic  time. 
But  when  r  is  known,  by  eliminating  the  effect  of  the  disturbing 
forces,  we  may  determine  with  accuracy  the  value  of  the  semi-trans- 
verse axis  a  at  each  epoch,  and,  from  this  and  the  observed  places, 
the  other  elements  of  the  orbit  according  to  the  process  already 
explained. 

Let  HQ  be  the  mean  daily  motion  at  the  first  epoch,  and  we  shall 
have 


*<=**, 


in  which  n  denotes  the  semi-circumference  of  a  circle  whose  radius  is 
unity.     Hence  we  obtain 

*  (220) 


by  means  of  which  to  determine  //0.  Then,  to  find  the  mean  daily 
motion  /*  at  the  instant  of  the  second  return  to  the  perihelion,  we 
have 


the  integral  being  taken  between  the  limits  0  and  r.  The  provisional 
value  of  the  mean  motion  as  given  by  the  observed  interval  r  will  be 
sufficiently  accurate  for  the  calculation  of  the  variations  of  M  and  ju 
during  this  interval.  The  semi-transverse  axis  will  now  be  derived 
by  means  of  the  formula 

,/F 


VARIATION   OF   CONSTANTS.  527 

from  the  values  of  //  for  the  two  epochs.  Let  r'  denote  the  interval 
which  must  elapse  before  the  next  succeeding  perihelion  passage  of 
the  comet,  and  we  have 

2r  _     T'   i    CdM  M 

and  consequently 

'dM_. 

5   '  (222) 


the  integral  being  taken  between  the  limits  t  =  0,  corresponding  to 
the  beginning  of  the  interval,  and  t  =  Tf.     We  have,  therefore, 


for  the  change  of  the  periodic  time  due  to  the  action  of  the  disturb- 
ing forces. 

198.  The  calculation  of  the  values  of  the  components  R,  S,  and  Z 
of  the  disturbing  force  will  be  effected  by  means  of  the  formulae 
given  in  Art.  182.  It  will  be  observed,  however,  that  not  only  these 
components  of  the  disturbing  force,  but  also  their  coefficients  in  the 
expressions  for  the  differential  coefficients,  involve  the  variable  ele- 
ments, and  hence  the  perturbations  which  are  sought.  But  if  we 
consider  only  the  perturbations  of  the  first  order,  the  fundamental 
osculating  elements  may  be  employed  in  place  of  the  actual  variable 
elements,  and  whenever  the  perturbations  of  the  second  order  have  a 
sensible  influence,  the  elements  must  be  corrected  for  the  terms  of  the 
first  order  already  obtained.  Then,  commencing  the  integration  anew 
at  the  instant  to  which  the  corrected  elements  belong,  the  calculation 
may  be  continued  until  another  change  of  the  elements  becomes 
necessary.  The  several  quantities  required  in  the  computation  of  the 
forces  may  also  be  corrected  from  time  to  time  as  the  elements  are 
changed. 

The  frequency  with  which  the  elements  must  be  changed  in  ordei 
to  include  in  the  results  all  the  terms  which  have  a  sensible  influence 
in  the  determination  of  the  place  of  the  disturbed  body,  will  depend 
entirely  on  the  circumstances  of  each  particular  case.  In  the  case  of 
the  asteroid  planets  this  change  will  generally  be  required  only  after 
an  interval  of  about  a  year;  but  when  the  planet  approaches  very 
near  to  Jupiter,  the  interval  may  necessarily  be  much  shorter.  The 


528  THEORETICAL   ASTRONOMY. 

magnitude  of  the  resulting  values  of  the  perturbations  will  suggest 
the  necessity  of  correcting  the  elements  whenever  it  exists;  and  if 
we  apply  the  pioper  corrections  and  commence  anew  the  integration 
for  one  or  more  intervals  preceding  the  last  date  for  which  the  per- 
turbations of  the  first  order  have  been  found,  it  will  appear  at  once, 
by  a  comparison  of  the  results,  whether  the  elements  have  too  long 
been  regarded  as  constant. 

The  intervals  at  which  the  differential  coefficients  must  be  com- 
puted directly,  will  also  depend  on  the  relation  of  the  motion  of  the 
disturbing  body  to  that  of  the  disturbed  body;  and  although  the  in- 
terval may  be  greater  than  in  the  case  of  the  variations  of  the  co- 
ordinates which  require  an  indirect  calculation,  still  it  must  not  be  so 
large  that  the  places  of  both  the  disturbing  and  the  disturbed  body,  as 
well  as  the  values  of  the  several  functions  involved,  cannot  be  inter- 
polated with  the  requisite  accuracy  for  all  intermediate  dates.  In  the 
case  of  the  asteroid  planets  a  uniform  interval  of  about  forty  days  will 
generally  be  preferred;  but  in  the  case  of  the  comets,  which  rapidly 
approach  the  disturbing  body  and  then  again  rapidly  recede  from  it, 
the  magnitude  of  the  proper  interval  for  quadrature  will  be  very 
different  at  different  times,  and  the  necessity  of  shortening  the  inter- 
val, or  the  admissibility  of  extending  it,  will  be  indicated,  as  the 
numerical  calculation  progresses,  by  the  manner  in  which  the  several 
functions  change  value. 

If  we  compute  the  forces  for  several  disturbing  bodies  by  using 
2R,  2S,  and  IZ  in  the  formulse  in  place  of  R,  S,  and  Z,  respect- 
ively, the  total  perturbations  due  to  the  combined  action  of  all  of 
these  bodies  may  be  computed  at  once.  But,  although  the  numerical 
process  is  thus  somewhat  abbreviated,  yet,  if  the  adopted  values  of 
the  masses  of  some  of  the  disturbing  bodies  are  uncertain,  and  it  is 
desired  subsequently  to  correct  the  results  by  means  of  corrected 
values  of  these  masses,  it  will  be  better  to  compute  the  perturbations 
due  to  each  disturbing  body  separately,  and,  since  a  large  part  of  the 
numerical  process  remains  unchanged,  the  additional  labor  will  not 
be  very  considerable,  especially  when,  for  some  of  the  disturbing 
bodies,  the  interval  of  quadrature  may  be  extended.  The  successive 
correction  of  the  elements  in  order  to  include  in  the  results  the  per- 
turbations due  to  the  higher  powers  of  the  masses,  must,  however, 
involve  the  perturbations  due  to  all  the  disturbing  bodies  considered. 

The  differential  coefficients  should  be  multiplied  by  the  interval  w, 
so  that  the  formulae  of  integration,  omitting  this  factor,  will  furnish 
directly  the  required  integrals;  and  whenever  a  change  of  the  inter- 


NUMERICAL   EXAMPLE. 


529 


val  is  introduced,  the  proper  caution  must  be  observed  in  regard  to 
the  process  of  integration.  The  quantity  s  =  206264".8  should  be 
introduced  into  the  formulae  in  such  a  manner  that  the  variations  of 
the  elements  which  are  expressed  in  angular  measure  will  be  obtained 
directly  in  seconds  of  arc;  and  the  variations  of  the  other  elements 
will  be  conveniently  determined  in  units  of  the  nth  decimal  place. 
Tt  should  be  observed,  also,  that  if  the  constants  of  integration  are 
put  equal  to  zero  at  the  beginning  of  the  integration,  the  integrals 
obtained  will  be  the  required  perturbations  of  the  elements. 

199.  EXAMPLE. — We  s.hall  now  illustrate  the  calculation  of  the 
perturbations  of  the  elements  by  a  numerical  example,  and  for  this 
purpose  we  shall  take  that  which  has  already  been  solved  by  the 
other  methods  which  have  been  given.  From  1864  Jan.  1.0  to  1865 
Jan.  15.0  the  perturbations  of  the  second  order  are  insensible,  and 
hence  during  the  entire  period  it  will  be  sufficient  to  use  the  values 
of  r,  v,  and  FJ  given  by  the  osculating  elements  for  1864  Jan.  1.0. 

The  calculation  of  the  forces  R,  S,  and  Z  is  effected  precisely  as 
already  illustrated  in  Art.  189,  and  from  the  results  there  given  we 
obtain  the  following  values  of  the  forces,  with  which  we  write  also 
the  values  of  EQ: — 


Berlin  Mean  Time.       40.R        40£ 

4QZ 

EQ 

1863 

Dec. 

12.0, 

+  0 

".0365 

+  0' 

'.0019  - 

f  0' 

'.00002 

355° 

26' 

8".2 

1864 

Jan. 

21.0, 

0 

.0356 

-0 

.0086 

0 

.00025 

8 

14 

57  .8 

March 

1.0, 

0 

.0315 

0 

.0182 

0 

.00047 

20 

57 

55  .1 

April 

10.0, 

0 

.0250 

0 

.0259 

0 

.00068 

33 

26 

47  .6 

May 

20.0, 

0 

.0169 

0 

.0314 

0 

.00087 

45 

35 

25  .3 

June 

29.0, 

-f  0 

.0079 

0 

.0343 

0 

.00101 

57 

20 

3  .8 

Aug. 

8.0, 

—  0 

.0011 

0 

.0349 

0 

.00112 

68 

39 

14  .6 

Sept. 

17.0, 

0 

.0099 

0 

.0333 

0 

.00117 

79 

33 

13  .1 

Oct. 

27.0, 

0 

.0179 

0 

.0301 

0 

.00116 

90 

3 

23  .2 

Dec. 

6.0, 

0 

.0252 

0 

.0253 

0 

.00108 

100 

11 

49  .1 

1865 

Jan. 

15.0, 

—  0 

.0317 

-0 

.0193  - 

f  o 

.00090 

110 

0 

54  .3 

"VVe  compute  the  values  of  the  required  differential  coefficients  by 
means  of  the  equations 


530 


THEORETICAL   ASTRONOMY. 


~~- 


—  -=  (a  cos  <p  sin  vR  +  a  cos  <p  (cos  v  -f-  cos  E)  S\ 
kvp 


3a/Jt     . 


d3M 
dt 


=     1     /  / 
&V\p  \  \ 


sm 


_  _ 


sm 


cos  _ 

' 


and  the  results  are  the  following: — 


40 


40 


dSir 


jUHWi 

"~dT 

dt 

VI 

dt 

dt 

10U 

u  —  •  — 

*y- 

~dT 

~dT 

1863  Dec. 

12.0, 

—  0".004 

—  0".001 

—  16".730 

+  0".022 

—  0".0790 

+  0".027 

+  11".092 

1864  Jan. 

21.0, 

0  .108 

0  .017 

17 

.255 

—  0  .992 

+  0 

.4524 

0 

.162 

11  .864 

March 

1.0, 

0  .302 

0  .026 

19 

.578 

1  .810 

0 

.9396 

0 

.863 

15  .381 

April 

10.0, 

0  .555 

0  .028 

22 

.986 

2  .294 

1 

.3321 

2 

.008 

20  .74(5 

May 

20.0, 

0  .822 

0  .022 

26 

.572 

2  .418 

1 

.6169 

3 

.492 

26  .898 

June 

29.0, 

1  .037 

—  0  .007 

29 

.271 

2  .228 

1 

.7750 

5 

.198 

32  .617 

Aug. 

8.0, 

1  .189 

+  0  .012 

30 

.698 

1  .829 

1 

.8196 

7 

.004 

37  .293 

Sept. 

17.0, 

1  .233 

0  .033 

30 

.500 

1  .406 

1 

.7591 

8 

.801 

40  .445 

Oct. 

27.0, 

1  .169 

0  .052 

28 

.953 

1  .055 

1 

.6206 

10 

.498 

42  .144 

Dec. 

6.0, 

1  .004 

0  .065 

26 

.498 

0  .902 

1 

.4074 

12 

.017 

42  .741 

1865  Jan. 

15.0, 

—  0  .742 

+  0  .066 

—  23 

.336 

—  1  .004 

+  1 

.1388 

+  13 

.292 

-f  42  .323 

The  values  thus  obtained  give,  by  means  of  the  formulae  for  integra- 
tion by  mechanical  quadrature,  the  following  perturbations  of  the 
elements : — 


Berlin  Mean  Time.               8Q 

Si 

Sir                     S(f>                        Sfi. 

ftjf 

1863  Dec. 

12.0, 

+  0' 

'.01 

—  0" 

.00 

+  8".43 

+  0" 

.12     - 

f  0".0007 

—  5".48 

1864  Jan. 

21.0, 

—  0 

.04 

0 

.01 

—  8 

.49 

—  0 

.38 

0 

.0040 

+  5   .72 

March 

1.0, 

0 

.24 

0 

.03 

26 

.78 

1 

.80 

0 

.0216 

19   .15 

April 

10.0, 

0 

.66 

0 

.06 

48 

.01 

3 

.88 

0 

.0502 

37    .11 

May 

20.0, 

1 

.35 

0 

.08 

72 

.82 

6 

.27 

0 

.0875 

60   .91 

June 

29.0, 

2 

.28 

0 

.10 

100 

.83 

8 

.61 

0 

.1299 

90   .73 

Aug. 

8.0, 

3 

.40 

0 

.09 

130 

.93 

10 

.65 

0 

.1751 

125   .79 

Sept. 

17.0, 

4 

.63 

0 

.07 

161 

.66 

12 

.26 

0 

.2200 

164   .79 

Oct. 

27.0, 

5 

.84 

—  0 

.03 

191 

.48 

13 

.48 

0 

.2624 

206   .19 

Dec. 

6.0, 

6 

.93 

4-0 

.03 

219 

.27 

14 

.44 

0 

.3004 

248    .72 

1865  Jan. 

15.0, 

y 

.81 

-f  0 

.10 

—  244 

.24 

—  15 

.37 

+  0 

.3323 

+  291    .33 

Applying  the  variations  of  the  elements  thus  obtained  to  the  oscu- 
lating elements  for  1864  Jan.  1.0,  as  given  in  Art.  166,  the  osculating 
elements  for  the  instant  1865  Jan.  15.0  are  found  to  be  the  following:  — 

Epoch  =  1865  Jan.  15.0  Berlin  mean  time. 
M=   99°  34'  48".81 
TT  =   44    13     7  .93 


i  =     4   36  52.  21 
?  =    11    15  35  .65 
log  a  =  0.3880283 
ft  =  928".8897. 


1860.0. 


NUMERICAL,   EXAMPLE.  531 

In  order  to  compare  the  results  thus  derived  with  the  nerturbations 
computed  by  the  other  methods  which  have  been  given,  let  us  com- 
pute the  heliocentric  longitude  and  latitude,  in  the  case  of  the  dis- 
turbed orbit,  for  the  date  1865  Jan.  15.0,  Berlin  mean  time.  Thus, 
by  means  of  the  new  elements,  we  find 

M=  99°  34'  48".81,         E=  110°  5'  14".15, 

logr=  0.4162182,  v  =  120  19  18  .01, 

I  =  164°  37'  59".04,         I  =  —  3   5  32  .54, 

agreeing  completely  with  the  results  already  obtained  by  the  other 
methods.  The  heliocentric  place  thus  found  is  referred  to  the  ecliptic 
and  mean  equinox  of  1860.0,  to  which  the  elements  ;r,  £2,  and  i  are 
referred ;  and  it  may  be  reduced  to  any  other  ecliptic  and  equinox  by 
means  of  the  usual  formulae.  Throughout  the  calculation  of  the  per- 
turbations it  will  be  convenient  to  adopt  a  fixed  equinox  and  ecliptic, 
the  results  being  subsequently  reduced  by  the  application  of  the  cor- 
rections for  precession  and  nutation. 

In  the  determination  of  dM,  if  we  denote  by  AM  the  value  which 

7  «  Ttr 

is  obtained  when  we  neglect  the  last  term  of  the  equation  for  —jr>  we 
shall  have 


which  form  is  equally  convenient  in  the  numerical  calculation.    Thus, 
for  1865  Jan.  15.0,  we  find 

AM  =  +  234".74, 

and  from  the  several  values  of  1600—^-  we  obtain,  for  the  same  datev 
by  means  of  the  formula  for  double  integration, 


Hence  we  derive 

dM  =  +  234".74  +  56".59  =  +  291".33, 

agreeing  with  the  result  already  obtained. 

If  we  compute  the  variation  of  the  mean  anomaly  at  the  epoch,  by 
means  of  equation  (209),  we  find,  in  the  case  under  consideration, 

dM  =  +  165".29, 


532  THEORETICAL   ASTKONOMY. 

and  since  the  place  of  the  body  in  the  case  of  the  instantaneous  orbit 
is  to  be  computed  precisely  as  if  the  planet  had  been  moving  con- 
stantly in  that  orbit,  we  have,  for  1865  Jan.  15.0, 


and  hence 

dM  =  3M0  +  (*  —  4,)  8?=  +  9.91"  M. 

The  error  of  this  result  is  —  0".23,  and  arises  chiefly  from  the  in- 
crease of  the  accidental  and  unavoidable  errors  of  the  numerical  cal- 
culation by  the  factor  t  —  109  which  appears  in  the  last  term  of  the 
equation  (209).  Hence  it  is  evident  that  it  will  always  be  preferable 
to  compute  the  variation  of  the  mean  anomaly  directly;  and  if  the 
variation  of  the  mean  anomaly  at  a  given  epoch  be  required,  it  may 
easily  be  found  from  dM  by  means  of  the  equation 


If  the  osculating  elements  of  one  of  the  asteroid  planets  are  thus 
determined  for  the  date  of  the  opposition  of  the  planet,  they  will 
suffice,  without  further  change,  to  compute  an  ephemeris  for  the  brief 
period  included  by  the  observations  in  the  vicinity  of  the  opposition, 
unless  the  disturbed  planet  shall  be  very  near  to  Jupiter,  in  which 
case  the  perturbations  during  the  period  included  by  the  ephemeris 
may  become  sensible.  The  variation  of  the  geocentric  place  of  the 
disturbed  body  arising  from  the  action  of  the  disturbing  forces,  may 
be  obtained  by  substituting  the  corresponding  variations  of  the  ele- 
ments in  the  differential  formulae  as  derived  from  the  equation  (1)2, 
whenever  the  terms  of  the  second  order  may  be  neglected.  It  should 
be  observed,  however,  that  if  we  substitute  the  value  of  dM  directly 
in  the  equations  for  the  variations  of  the  geocentric  co-ordinates,  the 
coefficient  of  d/j.  must  be  that  which  depends  solely  on  the  variation 
of  the  semi-transverse  axis.  But  when  the  coefficient  of  dp.  has  been 
computed  so  as  to  involve  the  effect  of  this  quantity  during  the  in- 
terval t  —  tw  the  value  of  dMQ  must  be  found  from  dM  and  substi- 
tuted in  the  equations. 

200.  It  will  be  observed  that,  on  account  of  the  divisor  e  in  the 
expressions  for  -£>  -TT-,  and  -^p  these  elements  will  be  subject  to  large 

perturbations  whenever  e  is  very  small,  although  the  absolute  effect 
on  the  heliocentric  place  of  the  disturbed  body  may  be  small  ;  and  on 


VARIATION  OF  CONSTANTS.  533 

account  of  the  divisor  sin  i  in  the  expression  for  —5—  the  variation 

of  &  will  be  large  whenever  i  is  very  small.  To  avoid  the  difficul- 
ties thus  encountered,  new  elements  must  be  introduced.  Thus,  in 
the  case  of  &,  let  us  put 

a"  =  sin  i  sin  &  ,  0"  =  sin  i  cos  &  ;  (224) 

then  we  shall  have 

da"  .  di    .  d£ 

—7—  =  sm  &  cos  i-j-  -f-  sin  t  cos  &—  7—  > 

at  ac  at 


-  cos  a  cos  t-  -  sm  *  sin 


Introducing  the  values  of  -^-  and  —  —  given  by  the  equations  (212), 

and  introducing  further  the  auxiliary  constants  a,  b,  A,  and  B  com- 
puted by  means  of  the  formulae  (94)x  with  respect  to  the  fundamental 
plane  to  which  &  and  i  are  referred,  we  obtain 

7      tl  -j 

rZsin  a  cos  (.4  +  w). 

(225) 

6  cos  (B  +  w), 


(1  -f  m) 

by  means  of  which  the  variations  of  a."  and  /9"  may  be  found.  If 
the  integrals  are  put  equal  to  zero  at  the  beginning  of  the  integration, 
the  values  of  da"  and  dp"  will  be  obtained,  so  that  we  shall  have 

sin  i  sin  &  =  sin  i0  sin  &0  -f-  £a", 
sin  i  cos  &  =  sin  i0  cos  &0  -f  &&'  * 
or 

sin  i  sin  (  &  —  £0)  =  cos  £0  #«"  —  sin  &0  <5y?", 

sin  i  cos  (&  —  $^0)  =  sin  i0  +  sin  ^0  ^a"  -J-  cos  ^0  dp",       (226) 

by  means  of  which  i  and  &  —  &0  may  be  found. 
In  the  case  of  #,  let  us  put 

y"  =  e  sin  /,  C"  =  e  cos  /,  (227  ) 

and  we  have 

de    .  dx 


d?'  de  dx 


534  THEORETICAL   ASTRONOMY. 

Substituting  for  -=-  and  -j-  the  values  given  by  the  equations  (203) 
and  (205),  and  reducing,  we  obtain 


-{-  m)  * 

-f-ersin/j/S), 

d"  1  I  (228) 

p  Sin  »  +  ^  ^  +  ^  +  ')  CQS 


ercos/j/Sl, 


by  means  of  which  the  values  of  dy"  and  ^f"  may  be  found.     Then 
we  shall  have 

e  sin  x  =  e0  sin  TTO  -f  &/', 
e  cos  /  =  e0  cos  TTO  -f-  <5C", 
or 

e  sin  (/  —  TTO)  =  cos  TTO  &/'  —  sin  TTO  <?:", 

e  cos  (/  —  TTO)  =  eQ  +  sin  TTO  5^"  -f  cos  TTO  5C",  (229) 

from  which  to  find  e  and  £.    If,  in  order  to  find  the  variation  of  TT,  we 
write  TT  instead  of  £  in  these  formulae,  the  terms  -f-  2ecos7rsin2^i—  7— 

7Q  «£ 

and  —  2esin^rsin2  &—JT  must  be  added  to  the  second  members  of 
(228),  respectively. 

201.  By  means  of  the  four  methods  which  we  have  developed  and 
illustrated,  the  special  perturbations  of  a  heavenly  body  may  be  de- 
termined with  entire  accuracy,  and  the  choice  of  the  particular  method 
will  depend  on  the  circumstances  of  the  case.  By  computing  the 
perturbations  of  the  elements,  correcting  these  elements  as  often  as 
may  be  required,  the  terms  depending  on  the  higher  powers  of  the 
masses  may  be  included,  and  no  indirect  calculation  becomes  necessary. 
The  frequent  correction  of  the  elements  will  also  render  insensible 
the  effect  of  whatever  uncertainty  remains  in  regard  to  their  true 
values.  But,  since  the  perturbations  of  the  elements  are  in  general 
much  greater  than  those  of  the  co-ordinates,  the  effect  of  the  terms 
of  the  second  order  will  be  much  greater  upon  the  values  of  the  ele- 
ments than  upon  those  of  the  co-ordinates.  Hence,  the  frequency 
with  which  a  change  of  the  elements  will  be  required  will  fully  com- 
pensate the  labor  of  the  indirect  part  of  the  calculation  in  the  case 
of  the  perturbations  of  the  co-ordinates. 


VARIATION   OF   CONSTANTS.  535 

The  determination  of  the  perturbations  of  the  polar  co-ordinates 
r,  Wj  and  z,  and  that  of  the  perturbations  dM,  v,  and  dzf,  are  effected 
with  almost  equal  facility,  especially  when  the  effect  of  the  disturb- 
ing forces  is  to  be  determined  for  a  long  interval  of  time.  If  the 
perturbations  are  required  only  for  a  brief  period,  it  will  be  prefer- 
able to  determine  dM,  v,  and  dz,  rather  than  dw,  Sr,  and  z,  since  the 
indirect  part  of  the  calculation  will  thus  be  effected  with  less  repe- 
tition. In  both  of  these  cases  the  values  of  the  perturbations  are 
generally  smaller  than  in  the  case  of  the  rectangular  co-ordinates,  and 
hence  they  are  less  affected  by  terms  of  the  second  order;  but  on 
account  of  the  simplicity  of  the  formulae,  even  when  we  include  the 
terms  depending  on  the  higher  powers  of  the  masses,  so  long  as  the 
magnitude  of  the  values  of  dx9  dy,  and  dz  is  not*  so  large  as  to 
render  troublesome  the  indirect  part  of  the  calculation,  the  method 
of  the  variation  of  rectangular  co-ordinates  may  be  advantageously 
employed  when  the  perturbations  are  to  be  determined  for  a  long 
period. 

By  whatever  method  the  perturbations  are  determined,  if  the  fun- 
damental osculating  elements  are  correct,  the  final  elements  of  the 
instantaneous  orbit  will  be  the  same.  But,  since  the  effect  of  the 
errors  of  the  elements  will  differ  in  degree  in  the  different  methods 
of  treating  the  problem,  if  these  elements  are  affected  with  small 
errors,  the  agreement  of  the  final  osculating  elements  obtained  by  the 
different  methods,  in  connection  with  the  corrections  derived  by  the 
comparison  of  observations,  may  not  be  complete. 

When  the  disturbed  body  approaches  very  near  to  a  disturbing 
planet,  the  magnitude  of  the  perturbations  will  be  such  as  to  enable 
us  by  means  of  accurate  observations  to  correct  the  adopted  value  of 
the  disturbing  mass.  In  this  case  the  perturbations,  computed  by 
means  of  either  of  the  methods  applicable,  must  be  converted  intc 
the  corresponding  perturbations  of  the  geocentric  spherical  co-ordi- 
nates. Let  the  variation  of  either  of  the  geocentric  co-ordinates 
arising  from  the  action  of  the  disturbing  planet  be  denoted  by  80; 
then,  if  we  suppose  the  correct  value  of  the  disturbing  mass  to  be 
1  -f-  n  times  the  assumed  value  used  in  computing  d6,  the  correspond- 
ing variation  of  the  geocentric  spherical  co-ordinate  will  be 

(1  -f  ri)  dff. 

The  value  dd  may  be  included  in  the  determination  of  the  difference 
between  computation  and  observation  in  the  formation  of  the  equa- 
tions of  condition  for  finding  the  corrections  to  be  applied  to  the  ele- 


536  THEORETICAL   ASTR3NOMY. 

ments;  and,  finally,  the  term  ndd  may  be  added  to  each  of  the  equa- 
tions of  condition,  so  that  we  thus  introduce  a  new  unknown  quantity 
n.  The  solution  of  all  the  equations  thus  formed,  by  the  method  of 
least  squares,  will  then  furnish  the  most  probable  values  of  the  cor- 
rections to  be  applied  to  the  adopted  elements,  and  also  the  value  of 
n,  by  means  of  which  a  corrected  value  of  the  mass  of  the  disturbing 
body  will  be  obtained. 

202.  If  the  determination  of  the  perturbations  of  a  heavenly  body 
required  that  all  the  disturbing  bodies  in  the  system  should  be  con- 
stantly considered,  the  labor  would  be  very  great.  But,  fortunately, 
it  so  happens  that  the  masses  of  many  of  the  planets  are  so  small  in 
comparison  with*  that  of  the  sun,  that  the  sphere  of  their  disturbing 
influence  is  very  much  restricted.  Thus,  in  the  determination  of  the 
perturbations  of  the  asteroid  planets,  only  the  action  of  Mars,  Jupi- 
ter, and  Saturn  need  be  considered ;  and  of  these  disturbing  planets 
Jupiter  exerts  the  principal  influence.  It  is  true,  however,  that,  on 
account  of  the  elongated  form  of  the  orbits  of  the  periodic  comets, 
they  may  at  different  times  be  sensibly  disturbed  by  each  of  the 
planets  of  the  system.  But  since  in  the  remote  parts  of  their  orbits 
they  are  very  distant  from  many  of  the  disturbing  planets,  the  deter- 
mination of  their  perturbations  will  then  be  much  facilitated  by  con- 
sidering them  as  revolving  around  the  common  centre  of  gravity  of 
the  sun  and  disturbing  planet.  When  the  motion  is  referred  to  the 
centre  of  the  sun,  the  disturbing  force  is  the  difference  of  the  direct 
action  of  the  disturbing  body  upon  the  disturbed  body  and  upon  the 
sun ;  and  in  the  case  of  those  disturbing  planets  whose  periodic  time 
is  short,  the  term  which  expresses  the  action  upon  the  sun  will  change 
value  so  rapidly  that  it  will  be  necessary  to  adopt  small  intervals  in 
the  direct  numerical  calculation.  But  when  we  refer  the  motion  to 
the  centre  of  gravity  of  the  system,  which  does  not  receive  any 
motion  in  virtue  of  the  mutual  attractions  of  the  bodies  which  com- 
pose the  system,  that  part  of  the  disturbing  force  which  expresses  the 
action  of  the  disturbing  planet  upon  the  sun  will  disappear,  and  the 
magnitude  of  the  disturbing  force  will  be  less  than  that  of  the  force 
which  disturbs  the  motion  of  the  comet  relative  to  the  sun,  so  that 
the  intervals  for  quadrature  may  be  greatly  extended.  It  will  be 
observed,  further,  that,  if  the  distance  of  the  comet  from  the  sun  is 
far  greater  than  the  distance  of  the  disturbing  body,  the  direct  action 
of  the  planet  upon  the  comet  becomes  so  small  that  its  effect  upon  the 
motion  will  be  quite  insignificant.  In  this  case  the  motion  of  the 


PERTURBATIONS   OF   COMETS.  537 

coinet  will  be  sensibly  the  same  as  the  pure  elliptic  motion  around 
the  common  centre  of  gravity  of  the  sun  and  disturbing  planet. 

In  order  to  exhibit  these  principles  more  clearly,  let  us  denote  by 
£,  37,  £,  the  co-ordinates  of  the  sun  referred  to  the  centre  of  gravity 
of  the  system ;  by  x0,  yw  z0,  the  co-ordinates  of  the  comet ;  and  by 
a?/,  y0',  z0',  the  co-ordinates  of  the  disturbing  planet  referred  to  the 
same  origin.  Let  x,  y,  z  be  the  co-ordinates  of  the  comet,  and 
x'9  y'j  z'  those  of  the  planet  referred  to  the  centre  of  the  sun;  then 
we  shall  have 


£  =  —  m'xj,         -TI  =  —  m'yj,          C  =  —  m'z0', 
and  hence 

a;  -=  x0  +  m'xj,        y  =  y0  +  m'yj,        z  =  z0  +  m'zj, 


From  these  we  derive 

mV  my  m'z9 

-IT^'      ^  "TT^1  "TT^'    (23 

The  equations  (15)j  are  now  easily  transformed  into  the  following:  — 


d\ 


_  (231) 


_    ,„,  ,      ,  _ 


which  completely  determine  the  motion  of  the  comet  about  the  com- 
mon centre  of  gravity  of  the  sun  and  planet.  The  second  members 
express  the  forces  which  disturb  the  pure  elliptic  motion;  and  it  is 
evident,  by  an  inspection  of  the  terms,  that  when  the  comet  is  remote 
from  both  the  planet  and  the  sun  these  forces  become  extremely 


538  THEORETICAL   ASTRONOMY. 

small.  If,  therefore,  we  compute  the  perturbations  of  the  motion 
relative  to  the  sun  as  far  as  to  the  point  at  which  the  second  members 
of  (231)  have  not  any  appreciable  influence  on  the  results,  it  will 
suffice  simply  to  convert  the  elements  which  refer  to  the  centre  of 
the  sun  into  those  relative  to  the  common  centre  of  gravity  of  the 
sun  and  disturbing  planet,  and  then  to  regard  the  motion  as  undis- 
turbed until  the  comet  again  approaches  so  near  that  the  direct  per- 
turbations must  be  considered,  at  which  point  the  motion  will  again 
be  referred  to  the  centre  of  the  sun. 

203.  The  reduction  of  the  elements  from  the  centre  of  gravity  of 
the  sun  to  the  common  centre  of  gravity  of  the  sun  and  the  disturb- 
ing planet,  may  be  easily  effected  by  means  of  the  variations  of  the 
rectangular  co-ordinates  and  of  the  corresponding  velocities.  To 
derive  the  co-ordinates  of  the  comet  referred  to  the  centre  of  gravity 
of  the  sun  and  planet,  it  is  only  necessary  to  add  to  the  heliocentric 
co-ordinates  the  co-ordinates  of  the  sun  referred  to  this  origin,  so 
that,  according  to  (230),  we  shall  have 


fe=-7T^>   (*»> 


and,  also, 


dy  m'         dtf 

~dt  =         1-fm'  '  "3T 
m'         «fr 
1  -f-  m'  '  eft  ' 

If,  therefore,  from  the  elements  of  the  orbit  of  the  disturbing  planet 
we  compute  the  auxiliary  constants  for  the  adopted  fundamental 
plane  by  means  of  the  equations  (94)x  or  (99)1?  and  also  V  and  Ur 
from 


VP' 


(er  sin  a/  +  sin  w')  =7'  sin  J7', 


l      m'  (er  cos  «/  +  cos  ti')  =  F  cos  U', 
' 


Vp' 

the  equations  (100)!  and  (49),  in  connection  with  (232)  and  (233), 
give 

9x  =  —      ™'r  ,  r'  sin  a'  sin  (A*  +  i*'),  (234^ 


PERTURBATIONS   OF   COMETS.  539 


dy  =  —  f  r'  sin  bf  sin  (£f  -f  u')t 

mf 

Sz  =  —  -T— r  r  sm  c  sm  ( (J  -f-  w') ; 

X   -J-   Wi 

*"  m' i'  +  U'),  (234) 


«  «"*   j,.,     .        ,  ^  ~,          TTf\ 

If  we  add  the  values  of  dx9  %,  £z,  3-j->  J-TT»  and  (5 -yr-  to  the  cor- 

w*         wt  dt 

responding  co-ordinates  and  velocities  of  the  comet  in  reference  to 
the  centre  of  gravity  of  the  sun,  the  results  will  give  the  co-ordinates 
and  velocities  of  the  comet  in  reference  to  the  common  centre  of 
gravity  of  the  sun  and  disturbing  planet,  and  from  these  the  new 
elements  of  the  orbit  may  be  determined  as  explained  in  Art.  168. 

The  time  at  which  the  elements  of  the  orbit  of  the  comet  may  be 
referred  to  the  common  centre  of  gravity  of  the  sun  and  planet,  can 
be  readily  estimated  in  the  actual  application  of  the  formulae,  by 
means  of  the  magnitude  of  the  disturbing  force.  In  the  case  of  Mer- 
cury as  the  disturbing  planet,  this  transformation  may  generally  be 
effected  when  the  radius-vector  of  the  comet  has  attained  the  value 
1.5,  and  in  the  case  of  Venus  when  it  has  the  value  2.5.  It  should 
be  remarked,  however,  that  the  distance  here  assigned  may  be  in- 
creased or  diminished  by  the  relative  position  of  the  bodies  in  their 
orbits.  The  motion  relative  to  the  common  centre  of  gravity  of 
the  sun  and  planet — disregarding  the  perturbations  produced  by  the 
other  planets,  which  should  be  considered  separately — may  then  be  re- 
garded as  undisturbed  until  the  comet  has  again  arrived  at  the  point 
at  which  the  motion  must  be  referred  to  the  centre  of  the  sun,  and  at 
which  the  perturbations  of  this  motion  by  the  planet  under  consider- 
ation must  be  determined.  The  reduction  to  the  centre  of  the  sun 
will  be  effected  by  means  of  the  values  obtained  from  (234),  when  the 
second  member  of  each  of  these  equations  is  taken  with  a  contrary 
sign. 

204.  In  the  cases  in  which  the  motion  of  the  comet  will  be  referred 
to  the  common  centre  of  gravity  of  the  sun  and  disturbing  planet, 
the  resulting  variations  of  the  co-ordinates  and  velocities  will  be  so 
smalJ  that  their  squares  and  products  may  be  neglected,  and,  there- 


540  THEORETICAL  ASTRONOMY. 

fore,  instead  of  using  the  complete  formulae  in  finding  the  new  ele- 
ments, it  will  suffice  to  employ  differential  formulae.  The  formulas 
(100),  give 

dx         .        .    f  .  dr    .        .  ,  A  dv 

-Tr  =  8iiia  sin  (  A  -f  u)  -%-  -f  r  sm  a  cos  (A  +  u)  -JT* 

Hj-  =  sin  b  sin  (B  +  u)  ^  +  r  sin  b  cos  (£  +  u)  ^-,        (235) 

(it  ut  (it 

dz         .        .      „        ,  dr  .  f  ~        .   dv 

-jr  =  sm  c  sm  (  C  +  w)  -57  +  r  sm  c  cos  (  (7  -f-  w)  -TI- 

at  at  at 

If  we  multiply  the  first  of  these  equations  by  3x,  the  second  by  dy} 
and  the  third  by  dz;  then  multiply  the  first  by  d—>  the  second  by 

7  7  Ctf 

^-^Y*  and  the  third  by  S-j->  and  put 
at  dt 


P=  sin  a  sin  (A  -{-u)  3x  -{-  sin  6  sin  (JB  -}-  w 

-f-  sin  c  sin  (  C  -}-  w)  ^2, 
Q  =  sin  a  cos  (  J.  +  w)  5«  -f  sin  6  cos  (JB  -f  w 

4-  sin  c  cos  ((7  -f-  u)  dz; 

P'  =  sin  a  sin  (4  +  i*)  *-  +  sin  6  sin  (B  +  ti)  d--       (236) 


g'  =  sin  a  cos  (J.  +  w)  d-^  +  sin  6  cos  (B  +  w)  J-~ 

-f-  sin  c  cos  (  C  +  u)  d—jr* 

we  shall  have,  observing  that  -—  =  -^=  e  sin  t?  and  that  -^  =  -^r- 


-dt0-dT--dr0-df-r-dtv^t 

From  the  equations 

dr          dx 


_ 

~  df  ^  dt'  ^  de' 


PERTUKBATIONS  OF  COMETS.  541 

we  get 


which  by  means  of  (237)  become 

'+!£«+«v      (238) 

=-=  esinvP'H-  ^  O^ 
Xf> 

From  the  equation 

V-W-TF  * 

we  get 


Substituting  the  values  given  by  (238),  observing  also  that  P=dr, 
this  becomes 

dk        dp  __  V*r  re*  sin2  v          esi 

T'h^—  ^J       "7T- 

and,  since 

F8  =  -(l  +  2ecosv+e«), 

we  obtain 

flr-.,          (239, 


by  means  of  which  the  variation  of  \/p  may  be  found. 
The  equation 

^.  =  2P_F» 
a         t 
gives 


from  which  we  derive 


542  THEORETICAL   ASTRONOMY. 

from  which  the  new  value  of  the  semi-transverse  axis  a  may  be 
found.     To  find  dp  we  have 

1  3k 

dfj.  =  Ifiad — J-  /A— -,  (241 ") 

a          k 

or 


Next,  to  find  8e,  we  have,  from  p  =  a  (1  —  e2), 


«•*—£*!—    '<**  (243> 


or 

jocose          smv  „      smv~l/p'f   ,    ^P  , 
to  =  — -j —  P  H §  H r-^1  -P  H 77-  (cos  v  +  cos  E} 


The  equation  (12)2  gives 

(2  +  e  cos  i>)  to,  (245) 

v 


2 
a2  cos?  a'cos'p 


— 


and  from  —  =  1  +  e  cos  v  we  get 


e  sin  i;          re  sm  v  re  sin  -y 

Substituting  this  value  of  dv  in  (245),  and  reducing,  we  find 


(24<y 


/cot,       tan^ 
^ 


(247^ 


tan,  '  \     r 

from  which  to  derive  the  variation  of  the  mean  anomaly. 


205.  Let  us  now  denote  by  a",  y",  2''  the  heliocentric  co-ordinates 
of  the  comet  referred  to  a  system  in  which  the  plane  of  the  orbit  is 
the  fundamental  plane,  and  in  which  the  positive  axis  of  x  is  directed 
to  the  ascending  node  on  the  ecliptic.  Let  us  also  denote  by  xf,  y',  zf 
the  co-ordinates  referred  to  a  system  in  which  the  plane  of  the  ecliptic 
is  the  plane  of  xy,  and  in  which  the  positive  axis  of  x  is  directed  to 
the  vernal  equinox.  Then  we  shall  have 


PERTURBATIONS   OF   COMETS.  543 


y"  =  —  x'  sin  &  cos  i  -f-  y'  cos  &  cos  i  -j-  zf  sin  i, 
2"  =  xr  sin  &  sin  i  —  y'  cos  &  sin  i  -\-  z'  cos  i. 

If  we  transform  the  co-ordinates  still  further,  and  denote  by  x,  y>  2 
the  co-ordinates  referred  to  the  equator  or  to  any  other  plane  making 
the  angle  e  with  the  ecliptic,  the  positive  axis  of  x  being  directed  to 
the  point  from  which  longitudes  are  measured  in  this  plane;  and  if 
we  introduce  also  the  auxiliary  constants  a,  A,  6,  B,  &c.,  we  shall 
have 

dx"  =  sin  a  sin  A8x-\-  sin  b  sin  B  dy  -f  sin  c  sin  C  8z, 

dy"  =  sin  a  cos  A  ox  -f-  sin  b  cos  B  dy  -f-  sin  c  cos  C  dz,        (248) 

dz"  —  cos  a  dx  -\-  cos  6  fy  -f-  cos  c  <te. 

Multiplying  the  first  of  these  by  —  sin  u,  and  the  second  by  cos  M, 
adding  the  results,  and  introducing  Q  as  given  by  the  second  of 
equations  (236),  we  get 

cos  u  ty"  —  sin  u  dx"  =  Q. 

Substituting  for  dxff  and  dy"  the  values  given  by  the  equations  (73)j, 
the  result  is 

r  (*o  -f  %)  =  Q, 

and,  introducing  the  value  of  dv  given  by  (246),  we  obtain 

Q        cosv 
d%  =  —  --  -.  —  3e 

r       esmv          r*e  sm  v 

Substituting  further  for  de,  dr,  and  d(i/p)  the  values  already  ob- 
tained, and  reducing,  we  find 

_  sinv  p      cos!?  n       cos  vVp  p,       (p  +  r)  sin  v 
'•~-         ~~^ 


by  means  of  which  8%  may  be  found. 
If  we  put 

cos  a  dx  4-  cos  bdy  +  cos  c  dz  = 


2  sin  v     dk 

T^X1 


the  last  of  the  equations  (248)  gives 


644  THEORETICAL  ASTRONOMY. 

dz"  =  R;  (251) 

and  if  we  differentiate  the  equation 

dx    ,         ,  dy    ,  dz 

cos  a-j-  -f-  cos  o-^j-  -f  cos  c-jr  =  0, 

at  at  at 

which  exists  in  the  case  of  the  unchanged  elements,  we  shall  have 

0  =  cos  a  d-r-  -f-  cos  b  d-~  -f  cos  e  8-=- 
at  at  at 

dx    .  dy    .  dz    . 

-=T-  sin  a  oa -f-  sin  b  do rr  sm  c  Sc. 

at  at  at 

Substituting  for  da,  8b,  and  do  the  values  given  in  Art.  60,  observing 
that  de  =  0,  we  have 

0=JR'-|-I  -^- sin  a  sin  A  -{- -—smbsmB  -{-  -57  sine  sin  (7 1  sin  tfl& 

— I  -j-  sin  a  cos  A  -j-  -j-  sin  b  cos  B  -f  -rr  sin  e  cos  C 1 8i. 
\  at  at  at  / 

From  the  equations  (100)1?  observing  that  the  relations  between  the 
auxiliary  constants  are  not  changed  when  the  variable  u  is  put  equal 
to  zero,  or  equal  to  90°,  we  get 

sin*  a  sin2  A  -\-  sin2  b  sin2  B  -J-  sin2  c  sin2  (7=1, 
sin2  a  cos'  A  -j-  sin2  b  cos2  J2  +  sin2  c  cos2  (7=1, 

and  from  (235)  we  find 

sin*  a  sin  A  cos  A  -f  sin2  5  sin  BcosB  -\-  sin2  c  sin  CcosC=  0.     (254) 

Substituting  in  (252)  for  -^»  -^-»  and  —  the  values  given  by  the 
equations  (49),  and  reducing  by  means  of  (253)  and  (254),  we  get 

0  =  R  —  FsinU'sin  i  3&  —VcosU  9L  (255) 

Substituting  further  for  dz"  in  (251)  the  value  given  by  the  last  of 
the  equations  (73)2,  there  results 

0  =  R  -f-  r  cos  u  sin  i  d&  —  r  sin  u  <H.  (256) 

From  these  equations  we  derive,  by  elimination, 


PERTURBATIONS   OF   COMETS.  545 

ecosw-J-cosw  1        rsiuu     , 

:  -  -.  -  ±i  -\  --  =  •    —  -.  -  -  H  , 

psmi  kV         sin  i 


,.  e  sin  </>  -f  sin  w  _ 

ft  =  -  .#  -f-  —  —  — 


by  means  of  which  8Q  and  8i  may  be  found.     To  find  dot  and  d;r  we 
have 

&y  =  <?£  —  cos  id&  ,  d*  =  d%  +  2  sin2  £i<5&  ,  (258) 


d%  being  found  from  equation  (249). 

Neglecting  the  mass  of  the  comet  as  inappreciable  in  comparison 
with  that  of  the  sun,  the  attractive  force  which  acts  upon  the  comet 
in  the  case  of  the  undisturbed  motion  relative  to  the  sun  is  P,  but  in 
the  case  of  the  motion  relative  to  the  common  centre  of  gravity  of 
the  sun  and  planet  this  force  is  P  (1  +  m').  Hence  it  follows  that 
the  increment  of  this  force  will  be  m'&2,  and  we  shall  have 

dk 


by  means  of  which  the  value  of  this  factor,  which  is  required  in  the 
formula  for  ^(i/),  fl—  »  &c.,  may  be  found. 


206.  The  formula?  thus  derived  enable  us  to  effect  the  required 
transformation  of  the  elements.     In  the  first  place,  we  compute  the 

values  of  dx,  dy,  dz,  8—t  d—jr>  and  d-jr  by  means  of  the  formulae 

Ctv  Ctt>  Civ 

(234)  ;  then,  by  means  of  (236)  and  (250),  we  compute  P,  §,  R,  P'  , 
Qf,  and  jRr,  the  auxiliary  constants  a,  A,  &c.  being  determined  in 
reference  to  the  fundamental  plane  to  which  the  co-ordinates  are  re- 
ferred. When  the  fundamental  plane  is  the  plane  of  the  ecliptic,  or 
that  to  which  &  and  i  are  referred,  we  have 

sinc  =  sini,  0=0. 

The  algebraic  signs  of  cos  a,  cos  6,  and  cos  c,  as  indicated  by  the  equa- 
tions (101)17  must  be  carefully  attended  to.  The  formulae  for  the 
variations  of  the  elements  will  then  give  the  corrections  to  be  applied 
to  the  elements  of  the  orbit  relative  to  the  sun  in  order  to  obtain 
those  of  the  orbit  relative  to  the  common  centre  of  gravity  of  the 
sun  and  planet.  Whenever  the  elements  of  the  orbit  about  the  sun 
are  again  required,  the  corrections  will  be  determined  in  the  same 
manner,  but  will  be  applied  each  with  a  contrary  sign. 

35 


546  THEORETICAL   ASTRONOMY. 

Since  the  equations  have  been  derived  for  the  variations  of  more 
than  the  six  elements  usually  employed,  the  additional  formulae,  as 
well  as  those  which  give  different  relations  between  the  elements  em- 
ployed, may  be  used  to  check  the  numerical  calculation;  and  this 
proof  should  not  be  omitted.  It  is  obvious,  also,  that  these  differen- 
tial formulae  will  serve  to  convert  the  perturbations  of  the  rectangular 
co-ordinates  into  perturbations  of  the  elements,  whenever  the  terms 
of  the  second  order  may  be  neglected,  observing  that  in  this  case 
8k  —  0.  If  some  of  the  elements  considered  are  expressed  in  angular 
measure,  and  some  in  parts  of  other  units,  the  quantity  s  =  206264".8 
should  be  introduced,  in  the  numerical  application,  so  as  to  preserve 
the  homogeneity  of  the  formulae. 

When  the  motion  of  the  comet  is  regarded  as  undisturbed  about 
the  centre  of  gravity  of  the  system,  the  variations  of  the  elements  for 
the  instant  t  in  order  to  reduce  them  to  the  centre  of  gravity  of  the 
system,  added  algebraically  to  those  for  the  instant  t'  in  order  to 
reduce  them  again  to  the  centre  of  the  sun,  will  give  the  total  pertur- 
bations of  the  elements  of  the  orbit  relative  to  the  sun  during  the 
interval  tf  —  t.  It  should  be  observed,  however,  that  the  value  of 
dM  for  the  instant  t  should  be  reduced  to  that  for  the  instant  tf,  so 
that  the  total  variation  of  M  during  the  interval  t1  —  t  will  be 


In  this  manner,  by  considering  the  action  of  the  several  disturbing 
bodies  separately,  referring  the  motion  of  the  comet  to  the  common 
centre  of  gravity  of  the  sun  and  planet  whenever  it  may  subsequently 
be  regarded  as  undisturbed  about  this  point,  and  again  referring  it  to 
the  centre  of  the  sun  when  such  an  assumption  is  no  longer  admissi- 
ble, the  determination  of  the  perturbations  during  an  entire  revolu- 
tion of  the  comet  is  very  greatly  facilitated. 

207.  If  we  consider  the  position  and  dimensions  of  the  orbits  of 
the  comets,  it  will  at  once  appear  that  a  very  near  approach  of  some 
of  these  bodies  to  a  planet  may  often  happen,  and  that  when  they 
approach  very  near  some  of  the  large  planets  their  orbits  may  be 
entirely  changed.  It  is,  indeed,  certainly  known  that  the  orbits  of 
comets  have  been  thus  modified  by  a  near  approach  to  Jupiter,  and 
there  are  periodic  comets  now  known  which  will  be  eventually  thus 
acted  upon.  It  becomes  an  interesting  problem,  therefore,  to  con- 
sider the  formulse  applicable  to  this  special  case  in  which  the  ordinary 
methods  of  calculating  perturbations  cannot  be  applied. 


PEKTUKBATIONS   OF   COMETS.  547 

If  we  denote  by  xf,  yr,  z',  r',  the  co-ordinates  and  radius-vector  of 
the  planet  referred  to  the  centre  of  the  sun,  and  regard  its  motion 
relative  to  the  sun  as  disturbed  by  the  comet,  we  shall  have 

fflaf 


dP  r" 

§f  +  -     1^sm)^  =  mp(^-|),  (260) 

W  +  ~    ~7*~ 

Let  us  now  denote  by  £,  57,  £  the  co-ordinates  of  the  comet  referred 
to  the  centre  of  gravity  of  the  planet;  then  will 


Substituting  the  resulting  values  of  #',  yf,  zr  in  the  preceding  equa- 
tions, and  subtracting  these  from  the  corresponding  equations  (1)  for 
*-he  disturbed  motion  of  the  comet,  we  derive 


These  equations  express  the  motion  of  the  comet  relative  to  the  centre 
of  gravity  of  the  disturbing  planet;  and  when  the  comet  approaches 
Very  near  to  the  planet,  so  that  the  second  member  of  each  of  these 
equations  becomes  very  small  in  comparison  with  the  second  term 
of  the  first  member,  we  may  take,  for  a  first  approximation, 


(fy      k*  (m  -f-  m')  I  _  0  (262) 

dt*  "*  >* 


_A 


and,  since  —  *  —  ^  —  -  is  the  sum  of  the  attractive  force  of  the  planet 
on  the  comet  and  of  the  reciprocal  action  of  the  comet  on  the  planet, 


548  THEORETICAL  ASTRONOMY. 

these  equations,  being  of  the  same  form  as  those  for  the  undisturbed 
motion  of  the  comet  relative  to  the  sun,  show  that  when  the  action 
of  the  disturbing  planet  on  the  comet  exceeds  that  of  the  sun,  the 
result  of  the  first  approximation  to  the  motion  of  the  comet  is  that 
it  describes  a  conic  section  around  the  centre  of  gravity  of  the  planet. 
Further,  since  — xf,  — yf,  — zf  are  the  co-ordinates  of  the  sun  re- 
ferred to  the  centre  of  gravity  of  the  planet,  it  appears  that  the 
second  members  of  (261)  express  the  disturbing  force  of  the  sun  on 
the  comet  resolved  in  directions  parallel  to  the  co-ordinate  axes 
respectively.  Hence  when  a  comet  approaches  so  near  a  planet  that 
the  action  of  the  latter  upon  it  exceeds  that  of  the  sun,  its  motion 
will  be  in  a  conic  section  relatively  to  the  planet,  and  will  be  dis  - 
turbed  by  the  action  of  the  sun.  But  the  disturbing  action  of  the 
sun  is  the  difference  between  its  action  on  the  comet  and  on  the 
planet,  and  the  masses  of  the  larger  bodies  of  the  solar  system  are 
such  that  when  the  comet  is  equally  attracted  by  the  sun  and  by  the 
planet,  the  distances  of  the  comet  and  planet  from  the  sun  differ  so 
little  that  the  disturbing  force  of  the  sun  on  the  comet,  regarded  as 
describing  a  conic  section  about  the  planet,  will  be  extremely  small. 
Thus,  in  a  direction  parallel  to  the  co-ordinate  £  the  disturbing  force 
exercised  by  the  sun  is 


l*_ 
r'3 


fa    I       /•  o          I  '"    I       /j 

and  when  the  comet  approaches  very  near  the  planet  this  force  will 
be  extremely  small.  It  is  evident,  further,  that  the  action  of  the 
gun  regarded  as  the  disturbing  body  will  be  very  small  even  when 
its  direct  action  upon  the  comet  considerably  exceeds  that  of  the 
planet,  and,  therefore,  that  we  may  consider  the  orbit  of  the  comet  to 
be  a  conic  section  about  the  planet  and  disturbed  by  the  sun,  when  it 
is  actually  attracted  more  by  the  sun  than  by  the  planet. 

208.  In  order  to  show  more  clearly  that  the  disturbing  force  of  the 
sun  is  very  small  even  when  its  direct  action  on  the  comet  exceeds 
that  of  the  planet,  let  us  suppose  the  sun,  planet,  and  comet  to  be 
situated  on  the  same  straight  line,  in  which  case  the  disturbing  force 
of  the  sun  will  be  a  maximum  for  a  given  distance  of  the  comet  from 

£1 

the  planet.     Then  will  the  direct  action  of  the  sun  be  — ,  and  that 

m'Tc* 
of  the  planet  — ^--      The  disturbing;  action  of  the  sun  will  be 


PERTURBATIONS   OF   COMETS.  549 


which,  since  p  is  supposed  to  be  small  in  comparison  with  r,  may  btj 
put  equal  to 


and  hence  the  ratio  of  the  disturbing  action  of  the  sun  to  the  direct 
action  of  the  planet  on  the  comet  cannot  exceed 


If  the  comet  is  at  a  distance,  such  that  the  direct  action  of  the  sun  is 
equal  to  the  direct  action  of  the  planet,  we  have 

p*  =  m'r\ 

and  the  ratio  of  the  direct  action  of  the  sun  to  its  disturbing  action 
cannot  in  this  case  exceed  21/m'.  In  the  case  of  Jupiter  this  amounts 
to  only  0.06. 

So  long  as  p  is  small,  the  disturbing  action  of  the  planet  is  very 

m'k* 
nearly  — —  in  all  positions  of  the  comet  relative  to  the  planet,  and 

hence  the  ratio  of  the  disturbing  action  of  the  planet  to  the  direct 
action  of  the  sun  cannot  exceed 


At  the  point  for  which  the  value  of  p  corresponds  to  R  =  R'y  the 
comet,  sun,  and  planet  being  supposed  to  be  situated  in  the  same 
straight  line,  it  will  be  immaterial  whether  we  consider  the  sun  or 
the  planet  as  the  disturbing  body  ;  but  for  values  of  p  less  than  this 
R  will  be  less  than  Rf,  and  the  planet  must  be  regarded  as  the  con- 
trolling and  the  sun  as  the  disturbing  body.  The  supposition  that 
R  is  equal  to  Rr  gives 


and  therefore 

p  =  rS/X5.  (263; 

Hence  we  may  compute  the  perturbations  of  the  comet,  regarding 
the  planet  as  the  disturbing  body,  until  it  approaches  so  near  the 


550  THEORETICAL   ASTRONOMY. 

planet  that  p  has  the  value  given  by  this  equation,  after  which,  so 
long  as  p  does  not  exceed  the  value  here  assigned,  the  sun  must  be 
regarded  as  the  disturbing  body, 

If  (p  represents  the  angle  at  the  planet  between  the  sun  and  comet, 
the  disturbing  force  of  the  sun,  for  any  position  of  the  comet  near 
the  planet,  will  be  very  nearly 


cos 


and  when  this  angle  is  considerable,  the  disturbing  action  of  the  sun 
will  be  small  even  when  p  is  greater  than  rl/^m'2.  Hence  we  may 
commence  to  consider  the  sun  as  the  disturbing  body  even  before  the 
comet  reaches  the  point  for  which 


and,  since  the  ratio  of  the  disturbing  action  of  the  planet  to  the 
direct  action  of  the  sun  remains  nearly  the  same  for  all  values  of  ^, 
when  p  is  within  the  limits  here  assigned  the  sun  must  in  all  cases 
be  so  considered.  Corresponding  to  the  value  of  p  given  by  equation 
(263),  we  have 


and  in  the  case  of  a  near  approach  to  Jupiter  the  results  are 
P  =  0.054  r,  #  =  0.33. 

209.  In  the  actual  calculation  of  the  perturbations  of  any  particu- 
lar comet  when  very  near  a  large  planet,  it  will  be  easy  to  determine 
the  point  at  which  it  will  be  advantageous  to  commence  to  regard  the 
gun  as  the  disturbing  body;  and,  having  found  the  elements  of  the 
prbit  of  the  comet  relative  to  the  planet,  the  perturbations  of  these 
elements  or  of  the  co-ordinates  will  be  obtained  by  means  of  the 
formulse  already  derived,  the  necessary  distinctions  being  made  in  the 
notation.  When  the  planet  again  becomes  the  disturbing  body,  the 
elements  will  be  found  in  reference  to  the  sun;  and  thus  we  are 
enabled  to  trace  the  motion  of  the  comet  before  and  subsequent  to  it*s 
being  considered  as  subject  principally  to  the  planet.  In  the  case  of 
the  first  transformation,  the  co-ordinates  and  velocities  of  the  comet 
and  planet  in  reference  to  the  sun  being  determined  for  the  instant  at 
which  the  sun  is  regarded  as  ceasing  to  be  the  controlling  body,  we 
shall  have 


PERTURBATIONS   OF   COMETS.  551 


d£__dx__dx'  di)  _  dy        dif  dZ  __  dz        cM  , 

~dt~~~dt~~~dt'  ~dt~~~dt~~~dt*  ~dt"  ~dt~~~dt' 

and  from  £,  57,  £,  -^-,  -7-,  and  -7%  the  elements  of  the  orbit  of  the 
comet  about  the  planet  are  to  be  determined  precisely  as  the  elements 
in  reference  to  the  sun  are  found  from  x,  yy  z,  —  yp  —j-,  and  -^->  and 

as  explained  in  Art.  168.  Having  computed  the  perturbations  of 
the  motion  relative  to  the  planet  to  the  point  at  which  the  planet  is 
again  considered  as  the  disturbing  body,  it  only  remains  to,  find,  for 
the  corresponding  time,  the  co-ordinates  and  velocities  of  the  comet 
in  reference  to  the  centre  of  gravity  of  the  planet,  and  from  these  the 
co-ordinates  and  velocities  relative  to  the  centre  of  the  sun,  and  the 
elements  of  the  orbit  about  the  sun  may  be  determined.  As  the  in- 
terval of  time  during  which  the  sun  will  be  regarded  as  the  disturb- 
ing body  will  always  be  small,  it  will  be  most  convenient  to  compute 
the  perturbations  of  the  rectangular  co-ordinates,  in  which  case  the 

values  of  £,  y,  £,  -TJ->  -^->  and  -rr  will  be  obtained  directly,  and  then, 
having  found  the  corresponding  co-ordinates  xr,  y',  z'  and  velocities 

—  rr>  -Jr»  —  7T  of  the  planet  in  reference  to  the  sun.  we  have 
at     at     at 


___  _  _  —         \  —          I 

~dt  ~~  ~dt  ""  ~di'  dt  ~~  dt  "  dt  '  dt  "=  dt    h  dt  ' 

by  means  of  which  the  elements  of  the  orbit  relative  to  the  sun  will 
be  found.  If  it  is  not  considered  necessary  to  compute  rigorously 
the  path  of  the  comet  before  and  after  it  is  subject  principally  to  the 
action  of  the  planet,  but  simply  to  find  the  principal  effect  of  the 
action  of  the  planet  in  changing  its  elements,  it  will  be  sufficient, 
during  the  time  in  which  the  sun  is  regarded  as  the  disturbing  body, 
to  suppose  the  comet  to  move  in  an  undisturbed  orbit  about  the 
planet.  For  the  point  at  which  we  cease  to  regard  the  sun  as  the 
disturbing  body,  the  co-ordinates  and  velocities  of  the  comet  relative 
to  the  centre  of  gravity  of  the  planet  will  be  determined  from  the 
elements  of  the  orbit  in  reference  to  the  planet,  precisely  as  the  corre- 
sponding quantities  are  determined  in  the  case  of  the  motion  relative 
to  the  sun,  the  necessary  distinctions  being  made  in  the  notation. 


552  THEOEETICAL   ASTRONOMY. 

210.  The  results  obtained  from  the  observations  of  the  periodic 
comets  at  their  successive  returns  to  the  perihelion,  render  it  probable 
that  there  exists  in  space  a  resisting  medium  which  opposes  the  motion 
of  all  the  heavenly  bodies  in  their  orbits;  but  since  the  observations 
of  the  planets  do  not  exhibit  any  effect  of  such  a  resistance,  it  is  in- 
ferred that  the  density  of  the  ethereal  fluid  is  so  slight  that  it  can 
have  an  appreciable  effect  only  in  the  case  of  rare  and  attenuated 
bodies  like  the  comets.  If,  however,  we  adopt  the  hypothesis  of  a 
resisting  medium  in  space,  in  considering  the  motion  of  a  heavenly 
body  we  simply  introduce  a  new  disturbing  force  acting  in  the  direc- 
tion of  the  tangent  to  the  instantaneous  orbit,  and  in  a  sense  contrary 
to  that  of  the  motion.  The  amount  of  the  resistance  will  depend 
chiefly  on  the  density  of  the  ethereal  fluid  and  on  the  velocity  of  the 
body.  In  accordance  with  what  takes  place  within  the  limits  of  our 
observation,  we  may  assume  that  the  resistance,  in  a  medium  of  con- 
stant density,  is  proportional  to  the  square  of  the  velocity.  The 
density  of  the  fluid  may  be  assumed  to  diminish  as  the  distance  from 
the  sun  increases,  and  hence  it  may  be  expressed  as  a  function  of  the 
reciprocal  of  this  distance. 

Let  ds  be  the  element  of  the  path  of  the  body,  and  r  the  radius- 
vector;  then  will  the  resistance  oe 


K  being  a  constant  quantity  depending  on  the  nature  of  the  body, 
and  <p  I  —  I  the  density  of  the  ethereal  fluid  at  the  distance  r.  Since 

the  force  acts  only  in  the  plane  of  the  orbit,  the  elements  which  de- 
fine the  position  of  this  plane  will  not  be  changed,  and  hence  we  have 
only  to  determine  the  variations  of  the  elements  M,  e,  a,  and  £.  If 
we  denote  by  ^0  the  angle  which  the  tangent  makes  with  the  prolon- 
gation of  the  radius-  vector,  the  components  R  and  8  will  be  given  by 


and,  since 

>  = 

Vp 


Jc  Jct/p  ds 

V  cos  (f>  =  -7-  e  sin  v,  Fsin  ^0  =  -  -,  V  =  -rr> 

r 


\ve 


have 


EESISTING   MEDIUM   IN   SPACE.  553 

Substituting  these  values  of  R  and  8  in  the  equation  (205),  it  reduces  to 

=  —  2Ky  (  -  1  sin  v  ds. 


Now,  since 


we  have 


V=  --  (I  -f-  2e  cos  v  +  e, 
Vp 


ds  =  Vdt  =  —  (l  +  2e  cos  v 


and  hence 

'     X  \  f*    '  V^OOJ 

If  we  suppose  the  function 

2e  cos  v  -f-  e*)3, 


(  i  j  r3  (1 


the  value  of  which  is  always  positive,  to  be  developed  in  a  series 
arranged  in  reference  to  the  cosines  of  v  and  of  its  multiples,  so  that 
we  have 

•^  (  7  )  **  C1  +  2e  cos  v  +  erf  =  A  +  ^  cos  v  +  c  cos  2v  +  &c->  (267> 

in  which  A,  B,  &c.  are  positive  and  functions  of  e,  the  equation  (266) 

becomes 

2 
ed%  =  --  (  A  -j-  -B  cos  v  -f-  .  .  .  .)  sin  v  dv. 

Hence,  by  integrating,  we  derive 


+  ----  ),  (268) 

from  which  it  appears  that  /  is  subject  only  to  periodic  perturbations 
on  account  of  the  resisting  medium. 

In  a  similar  manner  it  may  be  shown  that  the  second  term  of  the 
second  member  of  equation  (210)  produces  only  periodic  terms  in  the 
value  of  dM,  so  that  if  we  seek  only  the  secular  perturbations  due  to 
the  action  of  the  ethereal  fluid,  the  first  and  second  terms  of  the 
second  member  of  (210)  will  not  be  considered,  and  only  the  secular 
perturbations  arising  from  the  variation  of  //  will  be  required. 

Let  us  next  consider  the  elements  a  and  e.     Substituting  in  the 


554  THEORETICAL   ASTRONOMY. 

equations  (198)  and  (202)  the  values  of  E  and  S  given  by  (265),  and 
reducing,  we  get 

2o»       /  1  \  2  „  ..  | , 

da  = r-  K<p \  —  ]ry  (I  -}-  2e  cos  v  +  ez)    av, 

»          \  r  I 

2     ,/l\  J  (269) 

de  = K<p  I  —  I  r2  (1  -f-  2e  cos  v  -\-  e*)  .  (e  -f-  cos  v)  dv. 

p  \  7*  / 

If  we  introduce  into  these  the  series  (267),  and  integrate,  it  will  be 
found  that,  in  addition  to  the  periodic  terms,  the  expressions  for  da 
and  de  contain  each  a  term  multiplied  by  v,  and  hence  increasing  with 
the  time.  It  is  to  be  observed,  further,  that  since  A  and  B  are  posi- 
tive, the  secular  variation  of  a,  and  also  that  of  e,  will  be  negative, 
and  hence  the  resisting  medium  acts  continuously  to  diminish  both 
the  mean  distance  and  the  eccentricity. 

211.  The  magnitude  of  the  disturbing  force  arising  from  the  action 
of  the  resisting  medium  is  so  small  that  the  periodic  terms  have  no 
sensible  influence  on  the  place  of  the  comet  during  the  period  in 
which  it  may  be  observed ;  and  hence,  since  the  effect  of  the  resist- 
ance will  be  exhibited  only  by  a  comparison  of  observations  made  at 
its  successive  returns  to  the  perihelion,  the  effect  of  the  planetary  per- 
turbations being  first  completely  eliminated,  it  is  only  necessary  to 
consider  the  secular  variations.  Further,  since  y  is  subject  only  to 
periodic  changes  in  virtue  of  the  action  of  the  resistance,  and  since 
the  mean  longitude  is  subjected  to  a  secular  change  only  through  /*, 
it  will  suffice  to  employ  the  formulae  for  dp  and  de  or  d<p.  The 
variations  of  these  elements  may  be  computed  most  conveniently  by 

mechanical  quadrature  from  given  values  of  -77  and  -37-  or  -77-,  al- 

dt  ut          ut 

though  their  values  for  one  complete  revolution  of  the  comet  may  be 
determined  directly,  the  values  of  the  coefficients  A  and  B  which 
appear  in  the  series  (267)  being  found  by  means  of  elliptic  functions. 
The  calculation  of  the  effect  of  the  resisting  medium  will  be  made  in 
connection  with  the  determination  of  the  planetary  perturbations,  so 
that  there  will  be  no  inconvenience  in  adding  to  the  results  the  term? 
depending  on  this  resistance.  Since 

dfi  3  p.     da  d<p  de 

~dt  =     ~  2"  ~a  'W 

the  equations  (269)  give,  putting  K= 


RESISTING   MEDIUM   IN   SPACE.  555 


dt  r  cos  y> 


n\ 

\  r  I 


(270) 


It  remains  now  to  make  an  assumption  in  regard  to  the  law  of  the 
density  of  the  resisting  medium.  In  the  case  of  Encke's  comet  it 
has  been  assumed  that 


and  this  hypothesis  gives  results  which  suffice  to  represent  the  obser- 
vations at  its  successive  returns  to  the  perihelion.  Substituting  for  F 
its  value  in  terms  of  r  and  a,  the  equations  (270)  thus  become 


dt  r'r          a 


-  OM  TT 

-—  —  —  —  LKr  \J 


a  COS       COS  E 


—  r 

dt  v* 


l    2  * 

1  --- 

\  r         a 


by  means  of  which  djy.  and  d<p  may  be  found  ;  and  from  any  given 
value  of  dp.  we  may  derive  the  corresponding  value  of  da.  The 
variation  of  M,  neglecting  the  periodic  terms  arising  from  the  first 
and  second  terms  of  the  second  member  of  equation  (210),  will  be 
given  by 


which  will  be  integrated  by  mechanical  quadrature  so  as  to  include 
the  interval  of  an  entire  revolution  of  the  comet.  The  quantity  U 
has  been  determined,  by  means  of  observations  of  Encke's  comet,  to  be 


This  value  may  be  corrected  by  introducing  a  term  in  the  equations 
of  condition  precisely  as  in  the  case  of  the  determination  of  the  cor- 
rection to  be  applied  to  the  mass  of  a  disturbing  planet.  Intro- 
ducing U  into  the  equation  (264),  and  adopting  the  hypothesis  that 

^i  —  J  =  —  ,  the  expression  for  the  action  of  the  ethereal  fluid  be- 
comes 


556  THEORETICAL   ASTRONOMY. 

Since  the  constant  U  depends  on  the  nature  of  the  comet,  the  value 
obtained  in  the  case  of  Encke's  comet  may  be  very  different  from 
that  in  the  case  of  another  comet.  Thus,  in  the  case  of  Faye's  comet 
the  value  has  been  found  to  be 

U  =  10.232' 

and  in  the  application  of  the  formulae  to  the  motion  of  any  particular 
body  it  will  be  necessary  to  make  an  independent  determination  of 
this  constant. 

212.  The  assumption  that  the  density  of  the  ethereal  fluid  varies 
inversely  as  the  square  of  the  distance  from  the  sun,  is  that  which 
appears  to  be  the  most  probable,  and  the  results  obtained  in  accord- 
ance therewith  seem  to  satisfy  the  data  furnished  by  observation.  It 
is  true,  however,  that  the  whole  subject  is  involved  in  great  uncer- 
tainty as  regards  the  nature  of  the  resisting  medium,  so  that  the 
results  obtained  by  means  of  any  assumed  law  of  density  are  not  to 
be  regarded  as  absolutely  correct. 

From  the  formulae  which  have  been  given,  it  appears  that,  whatever 
may  be  the  law  of  the  density  of  the  resisting  fluid,  the  mean  motion 
is  constantly  accelerated  and  the  eccentricity  diminished,  and  we  may 
determine,  by  means  of  observations  at  the  successive  appearances  of 
the  comet,  the  amount  of  these  secular  changes  independently  of  any 
assumption  in  regard  to  the  density  of  the  ether.  Let  x  denote  the 
variation  of  p  during  the  interval  r,  which  may  be  approximately  the 
time  of  one  revolution  of  the  comet,  and  let  y  denote  the  correspond- 
ing variation  of  <p  ;  then,  after  the  lapse  of  any  interval  t  —  T0,  we 
shall  have 

(272) 


and,  since  the  average  variation  of  p.  during  the  interval  t  —  TQ  is 


M=  MQ  +  AIO  (*  -  T0)  +     ~r°    x.  (27,r 

If  we  introduce  x  and  y  as  unknown  quantities  in  the  equations  of 
condition  for  the  correction  of  the  elements  by  means  of  the  differ- 
ences between  computation  and  observation,  the  secular  variations  of 
fi  and  <p  may  be  determined  in  connection  with  the  corrections  to  b< 


RESISTING  MEDIUM  IN  SPACE.  657 

applied  to  the  elements.  For  this  purpose  the  partial  differential  co- 
efficients of  the  geocentric  spherical  co-ordinates  with  respect  to  x 
and  y  must  be  determined.  Thus,  if  we  substitute  the  values  of  /*, 
<p,  and  M  given  by  (272)  and  (273)  in  the  equations  (12)2  and  (14)2, 
we  obtain 

dr  (t  —  Tny        2r     t  —  Tn 

—  =  a  tan  <p  sin  v s — — -5—  • °  s, 

dx  2r  3/Jt         T 

dv       a2  cos?     (t  —  TQy  dr  t  —  T0 

-j-  = = - — s — — >  -j-  =  —  a  cos  <p  cos  v -,   (274) 

dx  r2  2r  rfy  T 

2 


dv       I 

-!—  =1 
dy       \ 


-!— 

dy       \  cos  <p 


. 
h  tan  #>  cos  v  I  sin  v 


in  which  s  =  206264". 8,  /*  being  expressed  in  seconds  of  arc.  Com- 
bining the  results  thus  obtained  with  the  differential  coefficients  of 
the  geocentric  spherical  co-ordinates  with  respect  to  r  and  v,  as  indi- 
cated by  the  equations  (42)2,  we  obtain  the  required  coefficients  of  x 
and  y  to  be  introduced  into  the  equations  of  condition.  The  solution 
of  all  the  equations  of  condition  by  the  method  of  least  squares  will 
then  furnish  the  most  probable  values  of  y  and  #,  or  of  the  secular 
variations  of  the  eccentricity  and  mean  motion,  without  any  assump- 
tion being  made  in  rei  jrence  either  t )  the  density  of  the  ethereal  fluid 
or  to  the  modifications  of  the  resistance  on  account  of  the  changes  in 
the  form  and  dimensions  of  the  comet,  and  the  results  thus  derived 
may  be  employed  in  determining  the  values  of  J!f,  //,  and  <p  for  the 
subsequent  returns  of  the  comet  to  the  perihelion. 

In  all  the  cases  in  which  the  periodic  comets  have  been  observed 
sufficiently,  the  existence  of  these  secular  changes  of  the  elements 
seems  to  be  well  established ;  and  if  we  grant  that  they  arise  from  the 
resistance  of  an  ethereal  fluid,  the  total  obliteration  of  our  solar 
system  is  to  be  the  final  result.  The  fact  that  no  such  inequalities 
have  yet  been  detected  in  the  case  of  the  motion  of  any  of  the  planets, 
shows  simply  the  immensity  of  the  period  which  must  elapse  before 
the  final  catastrophe,  and  does  not  render  it  any  the  less  certain. 
Such,  indeed,  appear  to  be  the  present  indications  of  science  in  re- 
gard to  this  important  question;  but  it  is  by  no  means  impossible 
that,  as  in  at  least  one  similar  case  already,  the  operation  of  the 
simple  and  unique  law  of  gravitation  will  alone  completely  explain 
these  inequalities,  and  assign  a  limit  which  they  can  never  pass,  and 
thus  afford  a  sublime  proof  of  the  provident  care  of  the  OMNIPOTENT 
CREATOR. 


TABLES. 


559 


TABLE  I,    Angle  of  the  Vertical  and  Logarithm  of  the  Earth's  Kadius, 

i 


Argument  0  =  Geographical  Latitude. 


Compression  = 


299.15 


4 

*-*' 

Diff. 

logp 

Diff. 

* 

*-,' 

Diff. 

logp 

Diff. 

0    / 

0  0 
1   0 
2  0 
3  0 
4  0 

0   0.00 

o  24.02 
o  48.02 

I  11.95 
I  35.80 

n 

24.02 
24.00 
23.93 
23.85 

o.ooo  oooo 

9-999  9996 
9982 
9961 
993° 

4 
14 

21 

31 

o   / 
35  0 

10 
20 
30 

40 

10  48.25 
49.67 
50.98 
52.31 
53.62 

a 

•38 
•35 
•33 

9-999  5248 
5208 
5169 
5129 
5089 

4° 
39 

40 
40 

1   5  0 

1  59-54 

23.74 
23.58 

9891 

48 

50 

54.90 

!26 

5049 

40 
4° 

i   6  0 

7  0 
8  0 

2  23.12 
2  46-54 

3  9-76 

23.42 
23.22 

9-999  9843 
9786 
9721 

l\ 

36  0 

10 
20 

10  56.16 

57-41 

58.63 

.25 

.22 

9-999  5°°9 
4969 

4929 

40 
40 

9  0 
10  0 

3  32.74 
3  55.47 

22.73 

9648 
9566 

11 

30 

40 

10  59.82 

II   1.00 

.18 

4888 
4848 

41 

40 

11  0 

4  J7-92 

22.45 
22.14 

'->  f 
9476 

90 

99 

50 

2.15 

•'5 

•J3 

4807 

40 

12  0 
13  0 
14  0 
15  0 
16  0 
17  0 

4  40.06 

5  1-85 
5  23.28 

5  44-33 
6  4.95 
6  25.14 

21.79 

21.43 
21.05 
20.62 
20.19 
19.72 

9-999  9377 
9271 

9'57 
9°35 
8905 
8768 

106 
114 

122 

I30 

137 
144 

37  0 

10 
20 
30 
40 
50 

II  3.28 

4-39 
5-47 
6-54 
7-58 
8-59 

.11 

.08 

.07 
.04 

I.OI 

1.  00 

9-999  4767 
4726 
4686 
4645 
4604 
4563 

41 
40 

41 
41 
41 
4' 

18  0 
19  0 
20  0 
21  0 
22  0 
23  0 

6  44.86 
7  4.09 
7  22.80 
7  40.99 
7  58.61 
8  15.66 

19.23 

18.71 
18.19 
17.62 
17.05 
16.44 

9.999  8624 
8472 
8314 
8149 
7977 
7799 

165 
172 
I78 
185 

38  0 

10 
20 

30 
40 
50 

ii  9.59 
10.56 
11.51 

12.44 

13-34 
14.22 

0.97 
0.95 
0.93 

0.90 
0.88 
0.86 

9-999  45" 

4481 
4440 
4399 
4358 
43'7 

41 

41 
41 
41 

24  0 
25  0 
26  0 
27  0 

28  0 

8  32.10 
8  47.93 

9  3-12 
9  17.65 

9  3x-5° 

I5-83 
15.19 

14-53 
I3-85 

9-999  7614 
7424 
7228 
7027 
6820 

190 
196 
201 
207 

39  0 

10 
20 
30 

40 

ii  15.08 
15.92 
16.73 
17.52 
18.29 

0.84 
0.8  1 
0.79 
0.77 

9-999  4276 
4234 
4*93 
4152 
4110 

42 
42 

29  0 

9  44.66 

13.16 
12.46 

6608 

212 

216 

50 

19.04 

0.75 
0.72 

4069 

42 

30  0 

10 
20 
30 
40 
50 

9  57.12 
9  59-12 

10   I.  II 

3-°7 
5.02 
6.94 

2.00 
1-99 
1.96 
1.95 
1.92 
1.91 

9-999  6392 
6355 
6319 
6282 
6245 
6208 

P 

37 
37 
37 

37 

40  0 

10 
20 
30 
40 
50 

ii  19.76 
20.46 
21.13 
21.79 

22.42 
23.02 

0.70 
0.67 
0.66 
0.63 
0.60 

9.999  4027 
3985 
3944 
3902 
3860 

42 
41 
42 
42 
41 
42 

31  0 

10 
20 

30 
40 
50 

10  8.85 
10.73 
12.59 
14.44 
16.26 
1  8.06 

1.88 
1.86 
I.g5 

1.82 
1.80 
1.78 

9.999  6171 

6096 
6059 
6021 
5984 

I 

38 
38 

41  0 

10 
20 
30 
40 
50 

ii  23.61 
24.17 
24.70 
25.22 
25.71 
26.18 

0.56 

o-53 
0.52 
0.49 
0.47 
0.44 

9-999  3777 
3735 
3693 
3651 
3609 
3567 

42 
42 
42 
42 
42 
42 

32  0 

10 
20 
30 
40 
50 

10  19.84 

21.  60 

23-34 

25.05 
26.75 
28.43 

1.76 
1.74 
1.71 

1.70 
1.68 
1.65 

9-999  5946 

5908 

5870 
5832 
5794 
5755 

38 

38 
38 
38 
39 
38 

42  0 

10 
20 
30 

40 
50 

ii  26.62 
27.04 

27.44 
27.82 
28.17 
28.50 

0.42 
0.40 
0.38 
o-35 
o-33 
0.30 

9-999  3525 
3483 

3399 
3357 
33'5 

42 
42 
42 
42 
42 

33  0 

10 
20 
cO 
40 

10  30.08 

33-32 

34-9J 

36.48 

1.63 
1.61 
i-59 

i-57 

9-999  5717 
5678 
5640 
5601 

39 
38 
39 
39 

43  0 

10 
20 
30 
40 

ii  28.80 
29.08 

29-34 
29.58 
29.79 

0.28 
0.26 
0.24 

0.21 

9-999  3273 

3270 
3188 
3H6 

3*°4 

43 
42 
42 
42 

42 

50 

38.03 

"•55 

1.52 

5523 

39 
39 

50 

29.98 

o.i  6 

3062 

T-* 

43 

34  0 

10 
20 
30 
40 
50 

10  39-55 
41.06 
42.54 
44.00 

JIU 

1.51 

1.48 
1.46 

i-44 
1.42 

I.  -JO 

9-999  5484 

5445 
5406 

5367 
5327 
5288 

39 
39 
39 

40 

39 
4° 

44  0 

10 
10 
SO 
40 
50 

ii  30.14 
30.29 

30.41 
30.50 

3°-57 
30.62 

0.15 

0.12 
0.09 
0.07 
0.05 
0.03 

9-999  3OI9 
2977 
2935 
2892 
2850 
2808 

42 
42 
43 
42 
42 
42 

35  0 

10  48.25 

J  7 

9-999  5248 

45  0 

ii  30.65 

9.999  2766 

661 


TABLE  I,    Angle  of  the  Vertical  and  Logarithm  of  the  Earth's  Eadius, 


=  Geocentric  Latitude. 


P  =  Earth's  Eadius. 


0 

0-0' 

Diff. 

logp 

Diff. 

0 

0-0' 

Diff. 

logp 

Diff. 

O     / 

45  0 

10 
20 
30 
40 
50 

II  30.65 
30.65 

30.63, 
30.58 

30.42 

0.00 
0.02 
O.O5 
0.07 
0.09 
Oil 

9.999  2766 
2723 
2681 
2639 
2596 

43 

42 
42 

43 
42 

O     / 

55  0 

10 
20 

30 
40 
50 

10  49.74 

48.36 
46.97 

45-55 

2% 

II 

1.38 
1.39 
1.42 

1.44 
1.46 

9-999  °*75 
0235 
0195 
0153 
OIK 
0076 

40 
40 
40 

39 
40 

42 

1.49 

39 

46  0 

10 
20 
30 

40 
50 

II  30.31 
30.17 
30.01 
29.82 
29.61 
29.38 

0.14 

0.16 
0.19 

0.21 
0.23 
O.26 

9.999  2512 
2470 
2427 
2385 

2343 
2300 

42 

43 
42 
42 

43 
42 

56  0 

10 
20 
30 
40 
50 

10  41.16 

39-65 
38.13 
36.5! 
35-oJ 
33-41 

1.51 

1.52 

'I7 

1.  60 
1.61 

9-999  °°37 
9.998  9998 

9958 
9919 
9880 
9841 

39 

40 

39 
39 
39 
39 

4T  0 
10 
20 
30 

II  29.12 
28.85 
28.54 
28.22 

0.27 
0.31 
0.32 

9.999  225* 
2216 
2174 
2132 

42 
42 
42 

57  0 

10 
20 
30 

10  31.80 
30.16 
28.50 
26.83 

1.64 
1.66 
1.67 

9.998  9802 

9764 
9725 
9686 

38 
39 
39 

40 
50 

27.87 
27.50 

°-35 
0-37 
0.40 

2089 
2047 

43 
42 
42 

40 
50 

23.40 

1.70 
1.73 
1.74 

9648 
9610 

39 

48  0 

10 
20 

II  27.10 
26.69 
26.24 

0.41 
0.45 

9.999  2005 
1963 
1921 

42 
42 

58  0 

10 
20 

10  21.66 
19.90 
18.11 

1.76 

9.998  9571 
9533 
9495 

1 

30 
40 

50 

25.78 
25.29 
24.78 

0.40 
0.49 
0.51 
0.54 

1879 
1837 
'795 

42 
42 
42 
42 

30 
40 
50 

16.31 

14.48 
12.63 

I'll 

1.85 
1.86 

9457 
9419 
9382 

| 

49  0 

10 
20 
30 
40 
50 

ii  24.24 
23.69 
23.11 

22.50 
21.87 

21.22 

0.55 
0.58 
0.61 
0.63 
0.65 
0.67 

9-999  »753 
1711 

1669 
1627 
1586 
1544 

42 
42 
42 

42 
42 

59  0 

10 
20 
30 
40 
50 

10  10.77 
8.88 
6.97 
5.04 
3-o8 

IO   I.  II 

1.89 
1.91 
1.93 
1.96 
1.97 
1.99 

9.998  9344 
9307 
9269 
9232 

9158 

11 

37 
37 
37 
37 

50  0 

10 
20 
30 
40 
50 

II  20.55 
19.85 

18.39 
17.63 
16.84 

0.70 
0.72 
0.74 
0.76 
0.79 
0.82 

9.999  1502 
1460 
1419 
'377 
^335 
1294 

42 

42 
42 

4' 

42 

60  0 
61  0 
62  0 
63  0 
64  0 
65  0 

9  59-i* 
9  46.74 
9  33-65 
9  19.85 

9  5-36 
8  50.21 

12.38 
13.09 
13.80 
14.49 
15.15 
15.81 

9.998  9121 
8902 
8688 

8479 
8275 
8077 

219 
214 
209 

204 
198 

51  0 

10 
20 
30 
40 
50 

II  l6.02 

15.19 

14-33 
13-45 
12.55 

11.62 

0-.83 
0.86 

0.90 

°-93 
0.95 

9.999  1252 

I2II 

II70 
1128 
1087 
1046 

42 

41 
41 

66  0 
67  0 
68  0 
69  0 
70  0 
71  0 

8  34-40 
8  17-97 
8  0.92 

7  43-29 
7  25.08 
7  6.33 

16.43 
17.05 
17.63 
18.21 
18.75 
19.27 

9.998  7884 
7697 
75'7 
734* 
7174 
7013 

187 
1  80 

III 

161 
J54 

52  0 

10 
20 
30 

II  10.67 

9.70 
8.71 

7.69 

0.97 
0.99 

.02 

9-999  1005 
0963 
0922 
O88l 

42 

72  0 
73  0 

74  0 
75  0 

6  47.06 
6  27.28 
6  7.03 
5  46-33 

19.78 

20.25 
20.70 

9.998  6859 
6713 

6573 
6441 

146 
140 
132 

40 
50 

6.66 
5.60 

3 

.09 

0840 
0800 

40 
41 

76  0 

77  0 

5  3-67 

21.13 

2I-53 
21.90 

6317 
6201 

Til 

1  08 

53  0 

10 
20 
30 
40 
50 

ii  4.51 
3.40 
2.27 

II   1.  12 

10  59.94 

58.74 

.11 

•13 
.1C 

.18 

.20 
.22 

9-999  °759 
0718 
0677 
0637 
0596 
0556 

40 

4i 
40 

41 

78  0 
79  0 

80  0 
81  0 
82  0 
83  0 

4  41-77 
4  19-53 
3  56.96 
3  34-io 
3  10.98 
2  47.63 

22.24 
22.57 

22.86 

23.12 

23-35 
23.56 

9.998  6093 

5993 
5901 
5818 

5676 

100 

ii 

57 

54  0 

10 
20 
30 
40 
50 

10  57.52 

56.28 
55.02 

53-73 
52.42 
51.09 

.29 

•33 
•35 

9.999  0515 
0475 
°435 
°395 
°355 
03*5 

40 
40 

4° 
40 
40 
4° 

84  0 
85  0 
86  0 
87  0 
88  0 
89  0 

2  24.07 
2   0.33 
I  36.44 
I  12.43 

o  4.8.3! 

o  24.18 

23:89 
24.01 
24.09 
24.16 
24.18 

9.998  5619 

5570 
553° 
5498 
5476 
5463 

49 
40 

3* 

22 

5 

55  0 

10  49.74 

J  J 

9.999  0275 

90  0 

O   O.OO 

9-998  5458 

662 


TABLE  II. 

For  converting  intervals  of  Mean  Solar  Time  into  equivalent  intervals  of  Sidereal 


Hours. 

Minutes. 

Seconds. 

Decimals. 

Mean  T. 

Sidereal  Time. 

Mean  T. 

Sidereal  Time. 

lean  T. 

Sidereal  Time. 

Mean  T. 

Sidereal  Time. 

h 

A    m        s 

m 

m         s 

g 

1 

s 

s 

I 

I   o     9.856 

I 

I    0.164 

I 

1.003 

0.02 

0.02O 

2 

2   0    19.713 

2 

2  0.329 

2 

2.005 

0.04 

0.04.0 

3 

3  o  29.569 

3 

3   0-493 

3 

3.008 

0.06 

O.OOO 

4 

4  o  39.426 

4 

4  0.657 

4 

4.011 

0.08 

0.080 

c 

5  o  49.282 

5  0.821 

5 

5.014. 

0.10 

O.  IOO 

i 

6  o  59-139 

6 

6  0.986 

6 

6.016 

0.12 

O.I  20 

1 

7         8.995 
8        18.852 

I 

7     -15° 
8     -3H 

i 

7.019 

8.022 

O.IJ 

o.i  6 

0.140 
O.I  60 

9 

9       28.708 

9 

9     .478 

9 

9.025 

0.18 

0.180 

10 

10       38.565 

10 

10      643 

10 

10.027 

0.2O 

0.201 

ii 

n       48.421 

ii 

ii     .807 

ii 

I  1.030 

0.22 

0.221 

12 

12          58.278 

12 

12       .971 

12 

12.033 

0.24 

0.241 

'3 

«4 

13    2       8.134 
14    2    17.991 

14 

13    2.136 
I4    2.300 

13 

«4 

I3-036 
14.038 

0.26 
0.28 

0.26l 
0.28l 

i5 

15    2    27.847 

'5 

'5  2.464 

15.041 

0.3.0 

0.301 

16 

16  2  37.704 

if 

16  2.628 

1  6 

16.044 

O.J2 

0.321 

18 

17    2    47.560 

1  8  2  5,7.416 

18 

17  2.793 
18  2.957 

ii 

17.047 
18.049 

0-34 
0.36 

0.341 
0.361 

19 

19  3     7.273 

19 

19  3.121 

19 

19.052 

0.38 

0.381 

20 

20  3  17.129 

20 

20    3.285 

20 

20.055 

0.40 

0.401 

21 

21  3  26.986 

21 

21     3.450 

21 

21.057 

0.42 

0.421 

22 

22    3    36.842 

22 

22    3.614 

22 

22.000 

o-44 

0.441 

23 
24 

23  3  46.699 
*4  3  56.555 

24 

24  3-943 

23 
24 

24.066 

.        f  O 

0.46 
o.4S 

0.461 
0.481 

11 

25  4.107 
26  4-271 

11 

25  ooo 
26.071 

0.50 
0.52 

0.501 
0.521 

oJ 

27 

27  4-435 

27 

27.074 

o-54 

0.541 

Q    C 

28 

28  4.600 

28 

28.077 

0.56 

0.562 

EH   8 

29 
30 

29  4-764 
30  4.928 

29 

3° 

29.079 
30.082 

0.58 
0.60 

0.582 
0.602 

g   i 

31 

31   5.092 

31 

31.085 

0.62 

O.622 

rQ       fl 

32 

S2  5-257 

32.088 

0.64 

0.642 

OQ     bfi 

33 

33  5-421 

33 

33.090 

0.66 

O.662 

2  .5   o5 

34 

34  5-585 

34 

34-093 

0.68 

0.682 

•8  15  § 

35 

35  5-750 

35 

35.096 

0.70 

0.702 

S  fj 

S6 

36  5-9H 

36 

36.099 

0.72 

0.722 

ill 

37 

37  6.078 

37 

37.101 

0-74 

0.742 

UU    4)    2 

PJ     rfi       H 

38    ' 

38  6.242 

38 

38.104 

0.76 

0.762 

*  1  s 

39 

39  6.407 

39 

39.107 

0.78 

0.782 

40 

40  6.571 

40 

40.110 

0.80 

0.802 

H    a  "So 

41  6-735 

41.112 

0.82 

0.822 

*o  .5    a) 

42 

42  6.899 

42 

42.115 

0.84 

0.842 

Ii! 

43 
44 

43  7-064 
44  7.228 

43 
44 

43.118 
44.120 

086 

0.88 

0.862 
0.882 

ill 

O     i,      c3 

46 

45  7-392 
46  7-557 

Jl 

45.123 
46.126 

0.90 
0.92 

0.902 
0.923 

0       .£ 

47 

47  7-721 

47 

47.129 

0.94 

0.943 

Ill 

48 

48  7.885 

48 

48.131 

0.96 

0.963 

5-1       O 
O    *J3      ^ 

49 

49  8-049 

49 

49.134 

0.98 

0.983 

5° 

50  8.214 

5° 

50.137 

1.  00 

1.003 

3  *  i 

51  8.378 

51 

51.140 

<3    ®    • 

5* 

52  8.542 

52 

52.142 

j§    S 

53 

53  8.707 

53 

53-145 

.s  £ 

54 

54  8.871 

54 

54.148 

<2      ^ 

II 

55, 
57 

55  9-035 
56  9  '99 
57  9-364 

I? 

57 

55-I5I 
56.153 
57.156 

e 

58 

58  9.528 

58 

58.159 

M 

I9 
60 

59  9.692 
60  9.856 

59 
60 

59.162 
60.164 

563 


TABLE  III. 

For  converting  intervals  of  Sidereal  Time  into  equivalent  intervals  of  Mean  Solar  Time, 


Hours. 

Minutes. 

Seconds. 

Decimals. 

Sid.  T. 

Mean  Time. 

Sid.  T. 

Mean  Time. 

Sid.  T. 

Mean  Time. 

Sid.  T. 

Mean  Time. 

h 

Am    « 

m 

m    5 

s 

8 

s 

s 

I 

o  59  50.170 

I 

o  59.836 

I 

0.997 

O.02 

0.020 

a 

i  59  40.341 

a 

I  59.672 

2 

1.995 

0.04 

0.040 

3 

a  59  30.511 

3 

a  59.509 

3 

2.992 

O.O6 

0.060 

4 

3  59  20.682 

4 

3  59-345 

4 

3-989 

0.08 

0.080 

5 

4  59  IO-852 

5 

4  59.181 

5 

4.986 

O.IO 

O.I  00 

6 

5  59  i-oa3 

6 

5  59017 

6 

5.984 

0.12 

0.120 

7 

6  58  51.193 

7 

6  58.853 

7 

6.981 

0.14 

0.140 

8 

7  58  4i-363 

8 

7  58.689 

8 

7-978 

0.16 

0.160 

9 

8  58  31.534 

9 

8  58.526 

9 

8.975 

0.18 

0.180 

10 

9  58  21.704 

10 

9  58-362 

10 

9-973 

0.20 

0.199 

ii 

10  58  11.875 

ii 

10  58.198 

ii 

30.970 

0.22 

0.219 

la 

ji  5$  2.045 

la 

ii  58.034 

12 

11.967 

0.24 

0.239 

J3 

la  57  52.216 

'3 

12  57.870 

13 

12.964 

0.26 

0.259 

H 

13  57  42.386 

«4 

13  57.706 

H 

13.962 

0.28 

0.279 

II 

H  57  32-557 
15  57  22.727 

11 

*4  57-543 
15  57-379 

!i 

14-959 
I5-956 

0.30 
0.32 

0.299 
0.319 

17 

16  57  12.897 

17 

16  57.215 

17 

16.954 

0-34 

o-339 

18 

17  57  3.068 

18 

17  57.051 

18 

17.953 

0.36 

o-359 

19 

18  56  53.238 

'9 

18  56.887 

19 

18.948 

0.38 

0:379 

ao 

19  56  43-4°9 

20 

19  56.723 

20 

19.945 

0.40 

o-399 

ai 

20  56  33.579 

21 

20  56.560 

21 

20.943 

0.42 

0.419 

aa 

21  56  23.750 

22 

21  56.396 

22 

21.940 

o-44 

0.439 

a3 

22  56  13.920 

a3 

22  56.232 

23 

22.937 

0.46 

0,459 

24 

43  56  4-091 

24 

23  56.068 

24 

a3-934 

0.48 

0-479 

a5 

24  55.904 

25 

24.932 

0.50 

0.499 

26 

25  55.740 

26 

25.929 

0.52 

0.519 

C) 

a7 

26  55.577 

27 

26.926 

0.54 

0.539 

a  § 

a8 

27  55-4I3 

28 

27.924 

0.56 

0.558 

EH  § 

a9 

28  55.249 

29 

28.921 

0.58 

0-578 

83  73 

30 

29  55.085 

3° 

29.918 

0.60 

0-598 

'o  g 

31 

30  54.921 

3s 

30.915 

0.62 

0.618 

r£j 

3a 

31  54-758 

32 

3I-9I3 

0.64 

0.638 

o3  "OD 

33 

32  54.594 

3.3 

32.910 

0.66 

0.658 

S   C?  S 

34 

33  54-43° 

34 

33.907 

0.68 

0.678 

Vn  '•& 

3S 

34  54.266 

35 

34.904 

0.70 

0.698 

C  8  73 

36 

35  54.102 

36 

35-9oa 

0.72 

0.718 

i  II 

H 

36  53.938 
37  53-775 

P 

36.899 
37.896 

0.74 
0.76 

0-738 
0.758 

<D  rd  ^ 

39 

38  53-6ii 

39 

38.894 

0.78 

0.778 

1  5  I 

40 

39  53-447 

40 

39.891 

0.80 

0.798 

41 

40  53.283 

41 

40.888 

0.82 

0.818 

£-   03 

4a 

41  53.119 

4a 

41.885 

0.84 

0.838 

8*1 

43 

42  52.955 

43 

42.883 

086 

0.858 

*S  »  3 

44 

43  52.792 

44 

43.880 

0.88 

0.878 

1  a  | 

0  ||  ,3 

45 
46 

44  52.628 
45  52-464 

Ji 

44.877 
45.874 

0.90 
0.92 

0.898 
0.917 

o  II  73 

47 

46  52.300 

47 

46.872 

0.94 

0.937 

J  1!  '3 

48 

47  52.136 

48 

47.869 

0.96 

0.957 

g'g  S1 

49 

48  51.972 

49 

48.866 

0.98 

0.977 

**"  2  -S 

5° 

49  51.809 

5° 

49.863 

1.  00 

0.997 

f«—  '  *->  -5 

50  51.645 

50.861 

a  ~l~ 

5a 

51  51.481 

52 

51.858 

*.d 

53 

5a  S'-S1? 

53 

52.855 

.a^g 

54 

53  51.153 

54 

53-853 

3  1 

55 

54  50.990 

55 

54.850 

5  S3 

56 

55  50.826 

56 

55-847 

.2  8 

57 

56  50.662 

57 

56.844 

~  s 

58 

57  50-498 

58 

57.842 

*^ 

59 

58  50-334 

59 

58.839 

60 

59  S0-^0 

60 

59.836 

564 


TABLE  IV. 

For  converting  Hours,  Minutes,  and  Seconds  into  Decimals  of  a  Day. 


Hours. 

Decimal. 

Min. 

Decimal. 

Min. 

Decimal. 

Sec. 

Decimal. 

Sec. 

Decimal. 

1 

0.0416  + 

1 

.000694  + 

31 

.021527  + 

1 

.0000116 

31 

.0003588 

2 

•°833  + 

2 

.001388  + 

32 

.022222  + 

2 

.0000231 

32 

.0003704 

3 

.1250  + 

3 

.002083  + 

33 

.022916  + 

3 

.0000347 

33 

.0003819 

4 

.1666  + 

4 

.002777  + 

34 

.023611  + 

4 

.0000463 

34 

.0003935 

5 

.2083  + 

5 

.003472  + 

35 

.024305  + 

5 

.0000579 

35 

.0004051 

6 

.2500  + 

6 

.004166  + 

36 

.025000  + 

6 

.0000694 

36 

.0004167 

7 

0.2916  + 

7 

.004861  + 

37 

.025694  + 

7 

.0000810 

37 

.0004282 

8 

•3333  + 

8 

.005555  + 

38 

.026388+ 

8 

.0000925 

38 

.0004398 

9 

•375°  + 

9 

.006250  + 

39 

.027083  + 

9 

.0001042 

39 

.0004514 

10 

.4166  + 

10 

.006944  + 

40 

.027777+ 

10 

.0001157 

40 

.0004630 

11 

.4583  + 

11 

.007638  + 

41 

.028472  + 

11 

.0001273 

41 

.0004745 

12 

.5000  + 

12 

•008333  + 

42 

.029166  + 

12 

.0001389 

42 

.0004861 

13 

0.5416  + 

13 

.009027  + 

43 

.029861  + 

13 

.0001505 

43 

.0004977 

14 

.5833  + 

14 

.009722  + 

44 

•030555+ 

14 

.0001620 

44 

.0005093 

15 

.6250  + 

15 

.010416  + 

45 

.031250  + 

15 

.0001736 

45 

.0005208 

16 

.6666  + 

16 

.01  II  I  I  + 

46 

.031944  + 

16 

.0001852 

46 

.0005324 

17 

•7083  + 

17 

.011805  + 

47 

.032638+ 

17 

.0001968 

47 

.0005440 

18 

.7500  + 

18 

.012500  + 

48 

•033333+ 

18 

.0002083 

48 

.0005556 

19 

0.7916  + 

19 

.013194  + 

49 

.034027  + 

19 

.0002199 

49 

.0005671 

20 

•8333  + 

20 

.013888  + 

50 

.034722  + 

20 

.0002315 

50 

.0005787 

21 

.8750  + 

21 

.014583  + 

51 

.035416  + 

21 

.0002431 

51 

.0005903 

22 

.9166  + 

22 

.015277  + 

52 

.036111  + 

22 

.0002546 

52 

.0006019 

23 

0.9583  + 

23 

.015972  + 

53 

.036805  + 

23 

.0002662 

53 

.0006134 

24 

1.0000  + 

24 

.016666  + 

54 

.037500  + 

24 

.0002778 

54 

.0006250 

25 

.017361  + 

55 

.038194  + 

25 

.0002894 

55 

.0006366 

26 

.018055  + 

56 

.038888+ 

26 

.0003009 

56 

.0006481 

27 

.018750  + 

57 

•039583+ 

27 

.0003125 

57 

.0006597 

28 

.019444  + 

58 

.040277  + 

28 

.0003241 

58 

.0006713 

29 

.020138  + 

59 

.040972  + 

29 

.0003356 

59 

.0006829 

30 

.020833  H" 

60 

.041666  + 

30 

.0003472 

60 

.0006944 

The  sign  +,  appended  to  numbers  in  this  table,  signifies  that  the  last  figure  repeats  to  fcfinity 


TABLE  V. 

For  finding  the  number  of  Days  from  the  beginning  of  the  Y< 


Date. 

Com. 

Bi8. 

January  o.o 

0 

0 

February  o.o 
March  o.o 

31 

59 

11 

April  o.o 

90 

9' 

May  o.o 

120 

121 

June  o.o 

!5' 

152 

July  o.o 

181 

182 

August  o.o 

212 

2I3 

September  o.o 

243 

244 

October  o.o 

*73 

274 

November  o.o 

3°4 

3°5 

December  o.o 

334 

335 

565 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

0° 

1° 

2° 

3° 

M. 

Diff.  1". 

M. 

Difif.  1". 

M. 

Diff.  1". 

M. 

Diff.  1". 

0' 

1 

o.oooooo 
0.010908 

181.81 
181.81 

0.654532 
0.665442 

181.83 
181.83 

1.309263 
1.320178 

181.92 
181.92 

1.964393 
1.975316 

182.05 
182.06 

2 

0.021817 

181.81 

0.676352 

181.83 

1.331093 

181.92 

1.986240 

182.06 

3 

0.032725 

181.81 

0.687262 

181.84 

1.342008 

181.92 

1.997164 

182.06 

4 

0.043633 

181.81 

0.698172 

181.84 

1.352923 

181.92 

2.008087 

182.07 

5 

0.054542 

181.81 

0.709082 

181.84 

1.363839 

181.93 

2.019011 

182.07 

6 

0.065450 

181.81 

0.719993 

181.84 

1-374755 

181  93 

2.029936 

182.07 

7 

0.076358 

181.81 

0.730903 

181.84 

1.385670 

181  93 

2.040860 

182.07 

8 

0.087267 

181.81 

0.741813 

181.84 

1.396586 

181.93 

2.051785 

182.08 

9 

0.098175 

181.81 

0.752724 

181.84 

1.407502 

181.93 

2.062709 

182.08 

10 

0.109083 

181.81 

0.763634 

181.84 

1.418418 

181.94 

1.073634 

182.08 

11 

o.i  19992, 

181.81 

0-774545 

181.84 

1-4*9334 

181.94 

2.084559 

182.08 

12 

0.130900 

181.81 

0.785456 

181.84 

1.440251 

181.94 

2.095485 

182.09 

13 

0.141808 

181.81 

0.796366 

181.85 

1.451167 

181.94 

2.106410 

182.09 

14 

0.152717 

181.81 

0.807277 

181.85 

1.462083 

181.94 

2.117335 

182.09 

15 

0.163625 

181.81 

0.818188 

181.85 

1.473000 

181.95 

2.128261 

182.10 

16 

0-174534 

181.81 

0.829099 

181.85 

1.483917 

181.95 

2.139187 

182.10 

17 

0.185442 

181.81 

0.840010 

181.85 

1.494834 

181.95 

2.1501  14 

182.10 

18 
19 

0.196350 
0.207259 

181.81 
181.81 

0.850921 
0.861832 

181.85 
181.85 

1-505751 
1.516668 

181.95 
181.95 

2.161040 
2.171966 

181  ii 
182.11 

20 

0.218167 

181.81 

0.872743 

181.85 

1.527585 

181.96 

2.182894 

182.11 

21 

0.229076 

181.81 

0.883654 

181.86 

i-5385°3 

181.96 

2.193820 

182.12 

22 

0.239984 

181.81 

0.894566 

181.86 

1.549420 

181.96 

2.204747 

182.12 

23 

0.250893 

181.81 

0.905478 

181.86 

1.560338 

181.96 

2.215674 

182.12 

24 

0.261801 

181.81 

0.916389 

181.86 

1.571256 

181.96 

2.226602 

182.13 

25 
26 

0.272710 
0.283619 

181.81 
181.81 

0.927301 
0.938212 

181.86 
181.86 

1.582174 
1.593092 

181.97 
181.97 

2.237529 
2.248457 

182.13 
182.13 

27 

0.294527 

181.81 

0.949124 

181.86 

1.604010 

181.97 

*-*59385 

182.14 

28 

0.305436 

181.81 

0.960036 

181.86 

1.614928 

181.97 

2.270313 

182.14 

29 

0.316345 

181.81 

0.970948 

181.87 

1.625847 

181.97   2.281242 

182.14 

30 

0.327253 

181.81 

0.981860 

181.87 

1.636766 

181.98   2.292170 

182.14 

31 

0.338162 

181.81 

0.992772 

181.87 

1.647684 

181.98   2.303099 

182.15 

32 

0.349071 

181.81 

1.003684 

181.87 

1.658603 

181.98   2.314028 

182.15 

33 

0.359980 

181.81 

1.014596 

181.87 

1.669522 

181.98   2.324957 

182.15 

34 

0.370888 

181.81 

1.025509 

181.87 

1.680441 

181.99   *-335887 

182.16 

35 

0.381797 

181.81 

1.036421 

181.87 

1.691361 

181.99   2.346816 

182.16 

36 

0.392706 

181.81 

1.047334 

181.87 

1.702280 

181.99   2.357746 

182.16 

37 

0.403615 

181.81 

1.058246 

181.88 

1.713200 

181.99   2.368676 

182.17 

38 

0.414524 

181.82 

1.069159 

181.88 

1.724120 

182.00   2.379606 

182.17 

39 

°-4*5433 

181.82 

1.080072 

181.88 

1.735039 

182.00   2.390536 

182.17 

40 

0.436342 

181.82 

1.090985 

181.88 

1.745960 

182.00 

2.4^1467 

182.18 

41 

0.447251 

181.82 

1.101898 

181.88 

1.756880 

182.00 

2.411  ^98 

182.18 

42 

0.458160 

181.82 

1.112811 

181.89 

1.767800 

182.01 

2.425329 

182.18 

43 

0.469069 

181.82 

1.123724 

181.89 

1.778  21 

182.01 

2.434260 

182.19 

44 

0.479970 

181.82 

1.134637 

181.89 

1.789(41 

182.01 

2.445191 

182.19 

45 
46 

0.490888 
0.501797 

181.82 
181.82 

1.145550 
1.156464 

181.89 
181.89 

1.800562 
1.811483 

182.01 

182.02 

2.456123 
2.467055 

182.19 
182.20 

47 

0.512706 

181.82 

1.167377 

181.89 

1.822404 

182.02 

2477987 

182.20 

48 

0.523616 

181.82 

1.178291 

181.89 

1.833325 

182.02 

2.488919 

182.20 

49 

0-5345*5 

181.82 

1.189205 

181.90 

1.844247 

182.02 

2.499851 

182.21 

50 

0-545435 

181.82 

1.200119 

181.90 

1.855168 

182.03 

2.510784 

182.21 

51 

0.556344 

181.82 

1.211033 

181.90 

1.866090 

182.03 

2.521717 

182.22 

52 

0.567254 

181.82 

1.221947 

181.90 

1.877012 

182.04 

2.532650 

182.22 

53 

0.578163 

181.83 

1.232861 

181.90 

1.887934 

182.04 

*-543583 

182.22 

54 

0.589073 

181.83 

i-*43775 

181.91 

1.898856 

182.04 

2.554517 

182.23 

55 

°-599983 

181.83 

1.254689 

181.91 

1.909779 

182.04 

2.565450 

182.23 

56 

0.610892 

181.83 

1.265604 

181.91 

1.920701 

182.04 

2.576384 

182.23 

57 

0.621802 

181.83 

1.276518 

181.91 

1.931624 

182.05 

2.587319 

182.24 

58 

0.632712 

181.83 

1.287433 

181.91 

1.942547 

182.05 

2.598253 

182.24 

59 

0.643622 

181.83 

1.298348 

181.91 

1-953470 

182.05 

2.609187 

182.24 

60 

0.654532 

181.83 

1.309263 

181.92 

I-964393 

182.05 

2.620122 

182.25 

566 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

4° 

5° 

6° 

7° 

M. 

Diflf.  V. 

If, 

Diflf.  1". 

M. 

Diflf.  1". 

M. 

Diflf.  1". 

0 

2.620122 

182.25 

3.276651 

182.50 

3.934182 

182.80 

4.592917 

183.17 

1 

2.631057 

182.25 

3.287602 

182.50 

3-945J51 

182.81 

4.603907 

183.18 

2 

2.641993 

182.26 

3.298552 

182.51 

3.956119 

182.82 

4.614898 

183.18 

3 

2.652928 

182.26 

3-309503 

182.51 

3.967088 

182.82 

4.625889 

183.19 

4 

2.663864 

182.26 

3.320454 

182.52 

3.978058 

182.83 

4.636880 

183.19 

5 

2.674800 

182.27 

3-33I405 

182.52 

3.989028 

182.83 

4.647872 

183.20 

6 

2.685736 

182.27 

3.342356 

182.53 

3.999998 

182.84 

4.658864 

183.21 

7 

2.696672 

182.27 

3-353308 

182.53 

4.010968 

182.84 

4.669857 

183.21 

8 

2.707609 

182.28 

3.364260 

182.54 

4.021939 

182.85 

4.680850 

183.22 

9 

2.718546 

182.28 

3,375212 

182.54 

4.032911 

182.86 

4.691843 

183.23 

10 

2.729483 

182.29 

3-386165 

182.55 

4.043882 

182.86 

4-70*837 

183.24 

11 

2.740420 

182.29 

3.397118 

182.55 

4.054854 

182.87 

4.713831 

183.24 

12 

2.751358 

182.29 

3.408071 

182.56 

4.065826 

182.87 

4.724826 

183.25 

13 
14 

2.762295 
2.773233 

182.30 
182.30 

3.419024 
3.429978 

182.56 
182.57 

4.076799 
4.087772 

182.88 
182.88 

4.735821 
4.746816 

183.25 
183.26 

15 

2.784172 

182.31 

3.440932 

182.57 

4.098745 

182.89 

4.757812 

183.27 

16 

2.795110 

182.31 

3.45x887 

182.58 

4.109718 

182.90 

4.768809 

183.27 

17 

2.806049 

182.31 

3.462841 

182.58 

4.120692 

182.90 

4.779805 

183.28 

18 

2.816988 

182.32 

3-473796 

182.59 

4.131667 

182.91 

4.790802 

183.28 

19 

2.827927 

182.32 

3.484752 

182.59 

4.14264! 

182.91 

4.801800 

183.29 

20 

2.838867 

182.33 

3.495707 

182.60 

4.153616 

182.92 

4.812797 

183.30 

21 

2.849806 

182.33 

3.506663 

182.60 

4.164592 

182.93 

4.823796 

183.31 

22 

2.860746 

182.33 

3.517619 

182.61 

4.175568 

182.93 

183.32 

23 

2.871686 

182.34 

3-5*8575 

182.61 

4.186544 

182.94 

4.845794 

183.32 

24 

2.882627 

182.34 

3.539532 

182.61 

4.197520 

182.94 

4.856793 

183-33 

25 

2.893567 

182.35 

3.550489 

182.62 

4.208497 

182.95 

4.867793 

l83-34 

26 

2.904508 

182.35 

3.56i447 

182.62 

4.219474 

182.95 

4.878793 

183.34 

27 

2.915449 

182.36 

3.572404 

182.63 

4.230451 

182.96 

4.889794 

183.35 

28 

2.926391 

182.36 

3.583362 

182.63 

4.241429 

182.97 

4.900795 

183.36 

29 

2.937332 

182.36 

3-5943*° 

182.64 

4.252408 

182.97 

4.911797 

183.36 

30 

2.948274 

182.37 

3.605279 

182.64 

4.263386 

182.98 

4.922799 

183-37 

31 

2.959217 

182.37 

3.616238 

182.65 

4.274365 

182.99 

4.933801 

183.38 

32 

2.970159 

182.37 

3.627197 

182.65 

4.285344 

182.99 

4.944804 

183.38 

33 

2.981102 

182.38 

3.638156 

182.66 

4.296324 

183.00 

4.955807 

183-39 

34 

2.992045 

182.38 

3.649116 

182.66 

4.307304 

183  oo 

4.966811 

183.40 

35 

3.002988 

182.39 

3.660076 

182.67 

4.318284 

183.01 

4.977815 

183.41 

36 

3.013931 

182.39 

3.671037 

182.68 

4.329265 

183.01 

4.988820 

183.41 

37 

3.024875 

182.39 

3.681997 

182.68 

4.340246 

183.02 

4.999825 

183.42 

38 
39 

3.035819 
3.046763 

182.40 
182.40 

3.692958 
3.703920 

182.69 
182.69 

4.351228 
4.362210 

183.03 
183.03 

5.010830 
5.021836 

183-43 
18  3-43 

40 

3.057707 

182.41 

3.714*81 

182.70 

4-373'9* 

183.04 

5.032842 

183.44 

41 

3.068652 

182.41 

3-7*5843 

182.70 

4-384T75 

183.05 

5.043849 

183.45 

42 
43 

3-079597 
3.090542 

182.42 
182.42 

3.736806 
3.747768 

182.71 
182.71 

4-395I58 
4.406141 

183.05 
183.06 

5.054856 
5.065864 

183.46 
183.46 

44 

3.101488 

182.43 

3-75873I 

182.72 

4.417125 

183.06 

5.076872 

183.47 

45 

1  46 

;  47 

48 

3.112433 

3.123379 
3-'343*5 

182.43 
182.44 
182.44 
182.44 

3.769694 

3.780658 
3.791622 
3.802586 

182.72 
182.72 
182.73 
182.74 

4.428109 

4-439093 
4.450078 
4.461064 

183.07 
183.08 
183.08 
183.09 

5.087880 
5.098889 
5.109898 
5.120908 

183.48 
185.48 
183.49 
183.50  i 

49 

3.156219 

182.45 

3.813551 

182.74 

4.472049 

183.10 

5.131918 

183.51 

50 

3.167166 

182.45 

3.824515 

182.75 

4-483035 

183.10 

5.142929 

183.51 

51 

3.178113 

182.46 

3.835481 

182.76 

4.494022 

183.11 

5-I5394° 

183.52 

52 

3.189061 

182.46 

3.846446 

182.76 

4.505008 

183.12 

5.164951 

183-53 

53 

3.200009 

182.47 

3.857412 

182.77 

4-5I5995 

183.12 

5-J75963 

183.54 

54 

3.210957  • 

182.47 

3.868378 

182.77 

4.526983 

183.13 

5.186975 

183.54 

55 

3.221905 

182.48 

3-879345 

182.78 

4-537971 

183.14 

5.197988 

!83.55 

56 

3,232854 

182.48 

3.890312 

182.78 

4.548959 

183.14 

5.209002 

183.56 

57 

3.243803 

182.49 

3.901279 

182.79 

4^59948 

183.15 

5.220015 

183.57 

58 

3.254752 

182.49 

3.912246 

182.79 

4-570937 

183.15 

5.231029 

*83-57 

59 

3.265702 

182.49 

3.923214 

182.80 

4.581927 

183.16 

5.242044 

183.58 

60 

3.276651 

182.50 

3.934182 

182.80 

4.592917 

183.17 

5-*53°59 

183.59 

567 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

8° 

9° 

10° 

11° 

M. 

Diff.  1". 

M. 

Diff.  1". 

M. 

Diff.  1". 

M. 

Diff.  1". 

0' 

5.253059 

183.59 

5.914815 

184.06 

6.578391 

184.60 

7-243997 

185.19 

1 

5.264075 

183.59 

5.925859 

184.07 

6.589467 

184.61 

7.255109 

185.20 

2 

5  275090 

183.60 

5.936904 

184.08 

6.600544 

184.62 

7.266222 

185.21 

3 

5.286107 

183.61 

5-947949 

184.09 

6.611622 

184.63 

7-277335 

185.22 

4 

5.297124 

183.62 

5.958995 

184.10 

6.622700 

184.64 

7.288449 

185.23 

5 

5.308141 

183.62 

5.970041 

184.11 

6.633778 

184.65 

7.299563 

185.25 

6 

5.319159 

183.63 

5.981087 

184.11 

6.644857 

18466 

7.310678 

185.26 

7 

5  33oi77 

183.64 

5.992134 

184.12 

6.655937 

184  67 

7.321793 

185.27 

8 

5.341195 

183.65 

6.003182 

184.13 

6.667017 

184.67 

7.332909 

185.28 

9 

5.352214 

183.66 

6.014230 

184.14 

6.678098 

184.68 

7.344026 

185.29 

10 

5.363234 

183.66 

6.025279 

184.15 

6.689179 

184.69 

7-355*44 

185.30 

11 

5.374254 

183.67 

6.036328 

184.16 

6.700261 

184.70 

7.366262 

185.31 

12 
13 

5-385275 
5.396296 

183.68 
183.69 

6.047378 
6.058428 

184.17 
184.18 

6.711343 
6.722426 

184.71 
184.72 

7.377381 
7.388500 

185.32 
185-33 

14 

S-WS1? 

183.69 

6.069479 

184.18 

6.733510 

184.73 

7.399620 

185-34 

15 

5-4l8339 

183.70 

6.080530 

184.19 

6-744594 

184.74 

7.410741 

185.35 

16 

5.429361 

183.71 

6.091582 

184.20 

6.755679 

184.75 

7.421862 

185.36 

17 

5.440384 

183.72 

6.102634 

184.21 

6.766764 

184.76 

7.432983 

185.37 

18 

5.451407 

183.73 

6.113687 

184.22 

6.777850 

184.77 

7.444106 

185.38 

19 

5.462431 

183-73 

6.124740 

184.23 

6.788937 

184.78 

7-455230 

185.39 

20 

5-473455 

183.74 

6-135794 

184.24 

6.800024 

184.79 

7-466354 

185.40 

21 

5.484480 

183.75 

6.146849 

184.25 

6.  811112 

184.80 

7.477478 

185.4! 

22 

5-4955°5 

183.75 

6.157904 

184.25 

6.822200 

184.81 

7.488603 

185.42 

23 
24 

5-506530 
5-5I7556 

18376 
183.77 

6.168959 
6.180015 

184.26 
184.27 

6.833289 
6.844378 

184.82 
184.83 

7.499729 
7.510855 

185.43 
185.44 

25 

5.528583 

183.78 

6.191072 

184.28 

6.855468 

184.84 

7.521982 

185.46 

26 
27 

5.539610 
5.550637 

183.79 
183.79 

6.202129 
6.213187 

184.29 

184.30 

6.866559 
6.877650 

184.85 
184.86 

7.533110 
7-544239 

185.47 
185.48 

28 

5.561665 

183.80 

6.224245 

184.31 

6.888742 

184.87 

7-555368 

185.49 

29 

5-572693 

183.81 

6.235304 

184.32 

6.899834 

184.88 

7.566497 

185.50 

30 

5.583722 

183.82 

6.246363 

184.32 

6.910927 

184.89 

7.577628 

185.51 

31 

5-594752 

183.8; 

6.257422 

184.33 

6.922021 

184.90 

7.588759 

185.52 

32 

5.605782 

183.8; 

6.268482 

184.34 

6.933115 

184.91 

7-599890 

185.53 

33 

5.616812 

183.84 

6-279543 

l84-35 

6.944210 

184.92 

7.611022 

185.54 

34 

5.627843 

183.85 

6.290605 

184.36 

6.955305 

184.93 

7.622155 

185.55 

35 

5.638874 

183  86 

6.301667 

184.37 

6.966401 

184.94 

7.633289 

185.57 

36 
37 

5.6499  6 
5.660938 

183.87 
183.87 

6.312729 
6.323792 

184.38 
184.39 

6.977498 
6.988595 

184.95 
184.96 

7.644423 
7.655558 

185.58 
185.59 

38 

5.671971 

183.88 

6-334855 

1  84.40 

6.999693 

184.97 

7.666694 

185.60 

39 

5.683004 

183.89 

6.345919 

184.41 

7.010791 

184.98 

7.677830 

185.61 

40 

5.694038 

183.90 

6.356984 

184.41 

7.021890 

184.99 

7.688967 

185.62 

41 

5.705072 

183.91 

6.368049 

184.42 

7.032990 

185.00 

7.700104 

185.63 

42 

5.716106 

183.91 

6.379115 

184.43 

7.044090 

185.01 

7.711242 

185.64 

43 

5.727141 

183.92 

6.390181 

1  84.44 

7.055191 

185.02 

7.722381 

185.65 

44 

5.738177 

183-93 

6.401248 

184.45 

7.066292 

185.03 

7-733521 

185.66 

45 

5.749213 

183.94 

6.412315 

184.46 

7-077394 

185.04 

7.744661 

185.68 

46 

5.760250 

18395 

6.423383 

184.47 

7.088497 

185.05 

7.755802 

185.69 

47 

5.771287 

183.96 

6.434451 

184.48 

7.099600 

185.06 

7.766943 

185.70 

48 

5.782325 

183.96 

6.445520 

1  84.49 

7.1  10704 

185.07 

7.778085 

185-71 

49 

5-793363 

183.97 

6.456590 

184.50 

7.121808 

185.08 

7.789228 

185.72 

50 
51 

5.804401 
5.815440 

183.98 
183.99 

6.467660 
6.478731 

184.51 
184.52 

7.132913 
7.144019 

185.09 
185.10 

7.800372 
7.811516 

185.73 

185.74 

52 

5.826480 

184.00 

6.489802 

184.52 

7.155125 

185.11 

7.822661 

185.75 

53 

5.837520 

184.01 

6.500874 

184-53 

7.166232 

185.12 

7.833807 

•fH 

54 

5.848561 

184.01 

6.511946 

184.54 

7.177340 

185.13 

7.844953 

185.78 

55 

5.859602 

184.02 

6.523019 

184.55 

7.188448 

185.14 

7.856100 

Jo5-z9 

56 

(5.870644 

184.03 

6.534092 

184.56 

7-199557 

185.15 

7.867247 

185.80 

57 

5.881686 

184.04 

6.545166 

184-57 

7.210666 

185.16 

7.878396 

185.  Si 

58 
59 

5.892728 
5.903771 

184.05 
184.06 

6.556241 
6.567316 

184.58 
184.59 

7221776 
7.232886 

185.17 
185.18 

7.889545 
7.900694 

185.82 
185.83 

60 

5.914815 

184.06 

6.578391 

184.60 

7.243997 

185.19 

7.911845 

185.84 

568 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelior  in  a  Paralolic  Orbit. 


V. 

12° 

13° 

14° 

15° 

M. 

Diff.  1". 

M. 

Diff.  1". 

M. 

Diff.  1". 

M. 

Diff.  1". 

0' 

7.911845 

185.84 

8.582146 

186.56 

9.255120 

187.33 

9.930984 

188.16 

1 

7.922995 

185.86 

8.593340 

186.57 

9.266360 

187.34 

9.942274 

188.18 

2 

7-934I47 

185.87 

8.604535 

186.58 

9.277601 

187.35 

9-953565 

188.19 

3 

7.945300 

185-88 

8.615730 

186.59 

9.288842 

187.37 

9.964857 

188.21 

4 

7.956453 

185.89 

8.626926 

186.61 

9.300085 

187.38 

9.976149 

188.22 

5 

7.967606 

185.90 

8.638123 

186.62 

9.311328 

187.40 

9.987443 

188.23 

6 

7.978761 

185.91 

8.649320 

186.63 

9.322572 

187.41 

9.998738 

188.25 

7 

7.989916 

185.92 

8.660518 

186.64 

9-3338I7 

187.42 

10.010033 

188.26 

8 

8.001072 

l85-93 

8.671717 

186.66 

9.345063 

187.44 

10.021329 

188.28 

9 

8.012228 

185-95 

8.682917 

186.67 

9.356310 

187.45 

10.032626 

188.29 

10 

8.023385 

185.96 

8.694117 

186.68 

9-367557 

187.46 

10.043924 

188.31 

11 

8.034543 

185.97 

8.705318 

186.69 

9.378805 

187.48 

10.055223 

188.32 

12 

8.045702 

185.98 

8.716520 

186.71 

9.390054 

187.49 

10.066523 

188.34 

13 

8.056861 

185.99 

8.727723 

186.72 

9.401304 

187.50 

10.077823 

188.35 

14 

8.068021 

186.00 

8.738927 

186.73 

9-41*555 

187.52 

10.089125 

188.37 

15 

8.079181 

186.02 

8.750131 

186.74 

9.423806 

187.53 

10.100427 

188.38 

16 

8.090343 

186.03 

8.761336 

186.76 

9.435058 

187.54 

10.1  1  1730 

'^•39 

17 

8.101505 

186.04 

8.772542 

186.77 

9.446311 

187.56 

10.123035 

188.41 

18 

8.112668 

186.05 

8.783748 

186.78 

9-457565 

187.57 

10.134340 

188.42 

19 

8.123831 

186.06 

8-794955 

186.79 

9.468820 

187.59 

10.145646 

188.44 

20 

8.134995 

186.07 

8.806163 

186.81 

9.480076 

187.60 

10.156952 

188.45 

21 

8.146160 

186.09 

8.817372 

186.82 

9.491332 

187.61 

10.168260 

188.47 

22 

8.157326 

186.10 

8.828582 

186.83 

9.502589 

187.63 

10.179568 

188.48 

23 

8.168492 

186.11 

8.839792 

186.84 

9.513847 

187.64 

10.190878 

188.50 

24 

8.179659 

186.12 

8.851003 

186.86 

9.525106 

187.65 

10.202188 

188.51 

25 

8.190826 

186.13 

8.862215 

186.87 

9.536366 

187.67 

10.213499 

188.53 

26 

8.201995 

186.15 

8.873427 

186.88 

9547626 

187.68 

10.224812 

188.54 

27 

8.213164 

186.16 

8.884641 

186.90 

9.558888 

187.70 

10.236125 

188.56 

28 

8.224334 

186.17 

8.895855 

186.91 

9.570150 

187.71 

10.247439 

X^-S7 

29 

8.235504 

186.18 

8.907070 

186.92 

9.581413 

187.72 

10.258753 

188.59 

30 
31 

8.246675 
8.257847 

186.19 
186.20 

8.918286 
8.929502 

186.93 
186.95 

9.592676 
9.603941 

187.74 
187.75 

10.270069 
10.281386 

188.60 
188.62 

32 

8  269020 

186.22 

8.940719 

186.96 

9.615207 

187.77 

10.292703 

188.63 

33 

8.280193 

186.23 

8-95i937 

186.97 

9.626473 

187.78 

10.304021 

188.65 

34 

8.291367 

186.24 

8.963156 

186.99 

9.637740 

187.79 

10.315341 

188.66 

35 

8.302542 

186.25 

8.974376 

187.00 

9.649008 

187.81 

10.326661 

188.68 

36 

8.313717 

186.26 

8.985596 

187.01 

9.660277 

187.82 

10.337982 

188.69 

37 
38 

8.324893 
8.336070 

186.28 
186.29 

8.996817 
9.008039 

187.02 
187.04 

9.671547 
9.682817 

187.84 
187.85 

10.349304 
10.360627 

188.71 
188.72 

39 

8.347248 

186.30 

9.019262 

187.05 

9.694088 

187.86 

10.371951 

188.74 

40 
41 

8.358426 
8.369605 

186.31 

186.32 

9.030485 
9.041709 

187.06 
187.08 

9.705361 
9.716634 

187.88 
187.89 

10.383275 
10.394601 

188.75 
188.77 

42 

8.380785 

186.34 

9.052934 

187.09 

9.727908 

187.91 

10.405927 

188.78 

43 
44 

8.391966 
8.403147 

186.35 
186.36 

9.064160 
9.075387 

187.10 
187.12 

9.739182 
9.750458 

187.92 
18793 

10.417255 
10.428583 

188.80 
188.81 

45 

8.414329 

186.37 

9.086614 

187-13 

9.761734 

187.95 

10.439912 

188.83 

I  46 

8.425512 

186.38 

9.097842 

187.14 

9.773012 

187.96 

10.451242 

188.84 

47 

8.436695 

186.40 

9.109071 

187.16 

9.784290 

187.98 

10.462573 

188.86 

48 
49 

8.447879 
8.459064 

186.41 
186.42 

9.120301 
9.131531 

187.17 
187.18 

9795569 
9.806849 

187.99 
188.00 

10.473905 
10.485238 

188.87 
188.89 

50 

8.470250 

186.43 

9.142763 

187.20 

9.818129 

188.02 

10.496572 

188.90 

51 
52 

8.481436 
8.492623 

186.45 
186.46 

9-J53995 
9.165228 

187.21 
187.22 

9.829410 
9.840693 

188.03 
188.05 

10.507907 
10.519242 

188.92 
188.93 

53 

8.503811 

186.47 

9.176462 

187.23 

9.851977 

188.06 

10.530579 

188.95 

54 

8.515000 

186.48 

9.187696 

187.25 

9.863261 

18808 

10.541916 

188.97 

55 

8.526189 

186.49 

9.198931 

187.26 

9.874546 

188.09 

10-553*55 

188.98 

56 

8-537379 

186.51 

9.210167 

187.27 

9.885832 

188.10 

10.564594 

189.00 

57 

8.548569 

186.52 

9.221404 

187.29 

9.897118 

188.12 

10-575934 

189.01 

58 
59 

8.559761 
8'570953 

186.53 
186.54 

9.232642 
9.243880 

187.30 
187.31 

9.908406 
9.919694 

188.13 
188.15 

10.587276 
10.598618 

189.03 
189.04 

60 

8.582146 

186.56 

9.255120 

187.33 

9.930984 

188.16 

10.609961 

189.06 

669 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

16° 

17° 

18° 

19° 

M. 

Diff.  I". 

M. 

Diff.  1". 

M. 

Diff.  1". 

M. 

Diff.  I". 

0' 

10.609961 

189.06 

11.292277 

190.02 

11.978162 

191.04 

12.667850 

192.13 

I 

2 

10.621305 
10.632649 

189.07 
189.09 

11.303679 
11.315082 

190.03 
190.05 

11.989625 

12.001089 

191.06 
191.08 

12.679379 
12.690908 

192.15 
192.17 

3 

10.643995 

189.10 

11.326485 

190.07 

12.012554 

191.09 

12.702439 

192.19 

4 

10.655342 

189.12 

11.337889 

190.08 

12.024021 

191.11 

12.713970 

192.21 

5 

10.666690 

189.14 

11.349295 

190.10 

12.035488 

191.13 

12.725503 

192.22 

6 

10.678038 

189.15 

11.360701 

190.12 

12.046956 

191.15 

12.737037 

192.24 

7 

10.689388 

189.17 

11.372109 

190.13 

12.058425 

191.16 

I2-748573 

192.26 

8 

10.700738 

189.18 

11.383517 

190.15 

12.069896 

191.18 

12.760109 

192.28 

9 

10.712090 

189.20 

11.394927 

190.17 

12.081367 

191.20 

12.771646 

192.30 

10 

10.723442 

189.21 

11.406337 

190.18 

12.092840 

191.22 

12.783185 

192.32 

11 
12 

10.734795 
10.746149 

189.23 
189.24 

11.417749 
11.429161 

190.20 
190.22 

12.104313 

12.115788 

191.24 
191.25 

12.794724 
12.806265 

192.34 
192.36 

13 

10.757505 

189.26 

11.440575 

190.23 

12.127264 

191.27 

12.817807 

14 

10.768861 

189.28 

11.451989 

190.25 

12.138741 

191.29 

12.829350 

192.39 

15 

10.780218 

189.29 

"•463405 

190.27 

12.150219 

191.31 

12.840894 

192.41 

16 

10.791576 

189.31 

11.474821 

190.28 

12.161698 

191.32 

12.852440 

192.43 

17 

10.802935 

189.32 

11.486239 

190.30 

12.173178 

191.34 

12.863986 

192.45 

18 

10.814295 

189.34 

11.497657 

190.32 

12.184659 

191.36 

12.875534 

192.47 

19 

10.825655 

I89-35 

11.509077 

190.33 

12.196141 

191.38 

12.887082 

192.49 

20 

10.837017 

189.37 

11.520497 

190.35 

12.207624 

191,/j.o 

12.898632 

192.51 

21 

10.848380 

189.39 

11.531919 

190.37 

12.219108 

191.41 

12.910183 

192.53 

22 

10.859744 

189.40 

"•543342 

190.39 

12.230594 

191.43 

12.921736 

192.55 

23 

10.871108 

189.42 

"•554765 

190.40 

12.242080 

191.45 

12.933289 

192.56 

24 

10.882474 

189.43 

11.566190 

190.42 

12.253568 

191.47 

12.944843 

192.58 

25 

10.893840 

189.45 

11.577616 

190.44 

12.265057 

191.49 

12.956399 

192.60 

26 

10.905208 

189.47 

1  1.589042 

190.45 

12.276546 

191.50 

12.967956 

192.62 

27 

10.916576 

189.48 

11.600470 

190.47 

12.288037 

191.52 

12.979514 

192.64 

28 

10.927946 

189.50 

11.611899 

190.49 

12.299529 

191.54 

12.991073 

192.66 

29 

10.939316 

189.51 

11.623328 

190.50 

12.311022 

191.56 

13.002633 

192.68 

30 

10.950687 

189-53 

11.634759 

190.52 

12.322516 

191.58 

13.014195 

192.70 

31 

10.962059 

189.55 

11.646191 

190.54 

12.334011 

191.60 

13.025757 

192.72 

32 

10973433 

189.56 

11.657624 

190.56 

12.345508 

191.61 

13.037321 

192.74 

33 
34 

10.984807 
10.996182 

189.58 
189.59 

11.669057 
11.680492 

190.57 
190.59 

12.357005 
12.368503 

191.63 
191.65 

13.048886 
13.060452 

192.76 
192.78 

35 

11.007558 

189-61 

11.691928 

190.61 

12.380003 

191.67 

13.072019 

192.80 

36 

11.018935 

189.63 

11.703365 

190.62 

12.391504 

191.69 

13.083587 

192.82 

37 

11.030313 

189.64 

11.714803 

190.64 

12.403006 

191.70 

13.095157 

192.83 

38 

11.041692 

189.66 

11.726242 

190.66 

12.414509 

191.72 

13.106727 

192.85 

39 

11.053072 

189.67 

11.737682 

190.68 

12.426013 

191.74 

13.118299 

192.87 

40 

11.064453 

189.69 

11.749123 

190.69 

12-43751? 

191.76 

13.129872 

192.89 

41 

11.075835 

189.71 

11.760565 

190.71 

12.449023 

191.78 

13.141446 

192.91 

42 

11.087218 

189.72 

11.772008 

190.73 

12.460531 

191.80 

13.153022 

192.93 

43 

11.098602 

189.74 

11.783452 

190.74 

12.472039 

191.81 

13.164598 

192.95 

44 

11.109987 

189.76 

11.794897 

190.76 

12.483548 

191.83 

13.176176 

192.97 

45 

11.121372 

189.77 

11.806344 

190.78 

12.495059 

191.85 

i3-l87755 

192.99 

46 

11.132759 

189.79 

11.817791 

190.80 

12.506571 

191.87 

i3-'99335 

193.01 

47 

11.144147 

189.80 

11.829239 

190.81 

12.518083 

191.89 

13.210916 

193.03 

48 

11.155536 

189.82 

11.840689 

190.83 

12.529597 

191.91 

13.222498 

193.05 

49 

11.166925 

189.84 

11.852139 

190.85 

12.541112 

191.93 

13.234082 

193.07 

50 

11.178316 

189.85 

11.863590 

190.87 

12.552628 

191.94 

13.245667 

193.09 

51 

11.189708 

189.87 

11.875043 

190.88 

12.564145 

191.96 

13.257253 

193.11 

52 

II.20IIOO 

189.89 

11.886496 

190.90 

12.575664 

191.98 

13.268840 

193.13 

53 

11.212494 

189.90 

11.897951 

190.92 

12.587183 

192.00 

13.280428 

193.15 

54 

11.223889 

189.92 

11.909407 

190.94 

12.598704 

192.02 

13.292017 

I93-X7 

55 

11.235284 

189.93 

11.920863 

190.95 

12.610225 

192.04 

13.303608 

193.19 

56 

11.246681 

189.95 

11.932321 

190.97 

12.621748 

192.06 

13.315200 

193.21 

57 

11.258078 

189.97 

11.943780 

190.99 

12.633272 

192.07 

13.326793 

193.23 

58 

11.269477 

189.98 

191.01 

12.644797 

192.09 

13-338387 

193.25 

!  59 

11.280876 

190.00 

11.966700 

191.02 

12.656323 

192.11 

13.349982 

193.27 

60 

11.292277 

190.02 

11.978162 

191.04 

12.667850 

192.13 

13.361579 

193-29 

570 


TABLE  VI. 

For  finding  tne  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


20° 

21° 

22° 

23° 

M. 

Diff.  1". 

K, 

Diff.  1". 

M. 

Diff.  1". 

M. 

Diff.  1". 

0' 

13.361579 

193.29 

14.059591 

194-51 

14.762133 

195.80 

15.469459 

197.17 

1 

13-373177 

193-3I 

14.071262 

194-53 

14.773882 

195-83 

15.481290 

197.19 

2 

13-384776 

193-33 

14.082935 

194-55 

14.785632 

195.85 

15.493122 

197.21 

3 

13.396376 

193-35 

14.094608 

194-57 

14.797384 

195-87 

15-504956 

197.24 

4 

13.407977 

193-37 

14.106283 

194.59 

14.809137 

195.89 

15-516791 

197.26 

5 

13.419580 

193-39 

14.117960 

194.61 

14.820891 

I95-9I 

15.528627 

197.28 

6 

13.431183 

193-4I 

14.129637 

194.64 

1483*647 

19594 

15.540465 

197-31   i 

7 

13.442788 

193-43 

14.141316 

194.66 

14.844403 

195.96 

15.552304 

197.33 

8 

13-454394 

193-45 

14.152996 

194.68 

14.856161 

195-98 

I5-564H4 

'97-35 

9 

13.466002 

193-47 

14.164677 

194.70 

14.867921 

196.00 

15.575986 

197.38 

10 

13.477610 

193-49 

14.176360 

194.72 

14.879682 

196.03 

15.587830 

197-4° 

11 

13.489220 

I93-5I 

14.188044 

19474 

14.891444 

196-05 

15-599675 

197-43 

12 

13.500831 

193-53 

i4-i997*9 

194.76 

14.903208 

196.07 

15.611521 

197-45 

13 

I  3.5  12443 

193-55 

14.211415 

194.78 

I4-914973 

196.09 

15.623369 

'97-47 

14 

13.524056 

19357 

14.223103 

194.81 

14.926739 

196.12 

15.635218 

197.50 

15 
16 

13.535671 
13.547287 

193-59 
193.61 

14.234792 
14.246482 

194.83 
194.85 

14.938506 
14.950275 

196.14 
196.16 

15.647068 
15.658920 

197.52 
197-54 

17 

13.558904 

193-63 

14.258174 

194.87 

14.962045 

196.18 

15.670773 

197-57 

18 

13.570522 

14.269867 

194-89 

14.973817 

196.20 

15.682628 

197-59 

19 

13.582141 

193.67 

14.281561 

194.91 

14.985590 

196.23 

I5-694484 

197.61 

20 

13.593762 

193-69 

14.293256 

194-93 

14-997365 

196.25 

15.706342 

197.64 

21 
22 

13.605383 
13.617006 

I93-7I 
193-73 

14.304953 
14.316651 

194.95 
194.98 

15.009140 
15  020917 

196.27 
196.30 

15.718201 
15.730061 

197.66 
197.69 

23 

13.628631 

193-75 

14.3*835° 

195.00 

15.032696 

196.32 

15.741923 

I97-71 

24 

13.640256 

193-77 

14.340050 

195.02 

15-044475 

196.34 

I5-753786 

197-73 

25 

13.651883 

193-79 

14.351752 

195.04 

15.056256 

196.36 

15.765651 

197.76 

26 

13-663511 

193.81 

14.363455 

195.06 

15.068039 

196.39 

15.777517 

197.78 

27 

13.675140 

193-83 

H-375'59 

195.08 

15.079823 

196.41 

I5-789385 

197.80 

28 
29 

13.686770 
13.698401 

I93-85 
I93-87 

14.386865 
14.398572 

195.10 

'95-13 

15.091608 
i5-i°3394 

196.43 
196.45 

15.801254 
15.813124 

197.83 
197-85 

30 

13.710034 

I93-89 

14.410280 

195.15 

15.115182 

196.48 

15.824996 

197.88 

31 

13.721668 

193.91 

14.421990 

195.17 

15.126971 

196.50 

15.836870 

197-9° 

32 

13-733303 

193-93 

14-433700 

I95-19 

15.1  38762 

196.52 

15-848744 

197.92 

33 

13.744940 

193-95 

14.445412 

195.21 

15.150554 

196.54 

15.860620 

197-95 

34 

13.756577 

193-97 

14.457126 

195.23 

15.162348 

196.57 

15.872498 

197-97 

35 
36 
37 

13.768216 
13.779856 
13.791498 

19399 
194.01 
194.03 

14.468841 
14.480557 
14.492274 

195.26 
195.28 
195-3° 

15.174142 
15.185938 
15.197736 

196.59 
196.61 
196.64 

r  ff 

I5-884377 
15.896258 
15.908140 

198.00 
198.02 
198.04 

38 

13.803140 

194-05 

14.503992 

195-3* 

15.209535 

196.66 

15.920023 

198.07 

39 

13.814784 

194.07 

14.515712 

195-34 

15.221335 

196.68 

15.931908 

198.09 

40 

13.826429 

194.09 

14.527434 

195-36 

15.233137 

196.70 

15-943794 

198.12 

41 

13.838075 

194.11 

14.539156 

195-39 

15.244940 

196.73 

15-955682 

198.14 

42 

13.849723 

194.14 

14.550880 

195.41 

15.256744 

196.75 

15.967571 

198.17 

43 

13.861372 

194.16 

14.562605 

195-43 

15.268550 

196.77 

15.979462 

198.19 

44 

13.873022 

194.18 

14-574331. 

195-45 

15.280357 

196.80 

15-991354 

198.21 

45" 
i  46 

47 
I  48 

13.884673 
13-8963*5 
13-907979 
13-919634 

194.20 
194.22 
194.24 
194.26 

14.586059 
14.597788 
14.609519 
14.621250 

195-47 
195.50 
195.52 
195-54 

15.292165 

15-303975 
15-315786 

15-3*7599 

196.82 
196.84 
196.87 
196.89 

16.003248 
16.015143 
16.027039 
16.038937 

198.24 
198.26 
198.29 
198.31 

49 

13.931290 

194.28 

14.632983 

I95-56 

I5-339413 

196.91 

16.050836 

198.34 

50 
51 
52 
53 
54 

13.942948 
13.954606 
13.966266 
13.977927 
13.989590 

194-30 
194-3* 
194-34 
194.36 

I94-38 

14.644718 
14.656453 
14.668190 
14.679929 
14.691668 

195.58 
195.60 
195-63 
I95-65 
195.67 

15.351228 

15-363045 
15.374863 
15.386683 
15.398504 

196.94 
196.96 
196.98 
197.00 
197.03 

16.062737 
16.074639 
16.086543 
16.098449 
16.110355 

198.36 
198.38 
198.41 

198-43 
198.46 

55 

14.001254 

194.41 

14.703409 

195-69 

15.410326 

197.05 

16.122263 

198.48 

56 

14.012919 

194-43 

14-715151 

195.71 

15.422150 

197.07 

10.134173 

198.51 

57 

58 
59 

14.024585 
14.036252 
14.047921 

194.45 
194.47 
194.49 

14.726895 
14.738640 
14.750386 

195-74 
195.76 
195.78 

15-433975 
15.445802 
15.457630 

197.10 
197.12 
197.14 

16.146084 
16.157997 
16.16991  1 

198-53 
198.56 
198.58 

60 

14.059591 

194.51 

14.762133 

195.80 

15-469459 

197.17 

16.181826 

198.60 

571 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

24° 

25° 

26° 

27° 

M. 

Diff.  1". 

M. 

Diff.  I". 

M. 

Diff.  1". 

M. 

Diff.  1". 

0' 

16.  181826 

198.60 

16.899499 

200.12 

17.622747 

201.70 

18.351847 

203.37 

1 

16.193743 

198.63 

16.911507 

200.14 

17.634850 

201.73 

18.364050 

203.40 

2 

16.205662 

198.65 

16.923516 

200.17 

17.646954 

201.76 

18.376255 

203.42 

3 

16.217582 

198.68 

16.935527 

200.19 

17.659060 

201.78 

18.388461 

203.45  j 

4 

16.229503 

198.70 

16.947539 

200.22 

17.671168 

201.  8l 

18.400669 

203.48 

5 

16.241426 

198.73 

16959553 

200.24 

17.683278 

201.84 

18.412879 

203.51 

6 

16.253350 

198.75 

16.971568 

200.27 

17.695389 

201.87 

18.425090 

203.54 

7 

16.265276 

198.78 

16.983585 

200.30 

17.707502 

201.89 

18.437303 

203.57 

8 

16.277204 

198.80 

16.995604 

200.32 

17.719616 

201.92 

18.449518 

203.59 

9 

16.289133 

198.83 

17.007624 

200.35 

17.731732 

201.95 

18.461735 

203.62 

10 

16.301063 

198.85 

17.019646 

200.37 

17.743850 

201.97 

18.473953 

203.65 

11 

16.312995 

198.88 

17.031669 

200.40 

17.755969 

202.00 

18.486173 

203.68 

12 

16.324928 

198.90 

17.043694 

200.43 

17.768090 

2O2.O3 

18498395 

203.71 

13 

16.336863 

198.93 

17.055720 

200.45 

17.78021  3 

202.06 

18.510618 

203.74 

14 

16.348799 

198.95 

17.067748 

200.48 

17.792337 

202.08 

18.522843 

203.77 

15 

16.360737 

198.97 

17.079777 

200.50 

17.804462 

202.11 

18.535070 

203.80 

16 

16.372676 

199.00 

17.091808 

200.53 

17.816590 

202.14 

18.547299 

203.82 

17 

16.384617 

199.02 

17.103841 

200.56 

17.828719 

202.17 

18.559529 

203.85 

18 

16.396559 

199.05 

17.115875 

200.58 

17.840850 

2O2.I9 

18.571761 

203.88 

19 

16.408503 

199.07 

17.127911 

200.61 

17.852982 

202.22 

18.583995 

203.91 

20 

16.420448 

199.10 

17.139948 

200.64 

17.865116 

2O2.25 

18.596230 

203.94 

21 

16.432395 

199.12 

17.151987 

200.66 

17.877252 

202.28 

18.608467 

203.97 

22 

16.444343 

199.15 

17.164028 

200.69 

17.889389 

202.30 

18.620706 

204.00 

23 

16.456292 

199.17 

17.176070 

200.71 

17.901528 

202-33 

18.632947 

204.03 

24 

16.468243 

199.20 

17.188114 

200.74 

17.913669 

202.36 

18.645190 

204.05 

25 

16.480196 

199.22 

17.200159 

200.77 

17.925811 

202.39 

18.657434 

204.08 

26 

16.492151 

199.25 

17.212206 

200.79 

J7-937955 

202.41 

18.669679 

204.11 

27 

16.504107 

199.27 

17.224254 

200.82 

17.950101 

202.44 

18.681927 

204.14 

28 

16.516064 

199.30 

17.236304 

200.85 

17.962248 

202.47 

18.694177 

204.17 

29 

16.528022 

'99-33 

17.248356 

200.87 

17-974397 

202.50 

18.706428 

204.20 

30 

16.539983 

199-35 

17.260409 

200.90 

17.986548 

202.52 

18.718680 

204.23 

31 

l6-55J945 

199.38 

17.272464 

200.93 

17.998700 

202.55 

18.730935 

204.26 

32 

16.563908 

199.40 

17.284520 

200.95 

18.010854 

202.58 

18.743191 

204.29 

33 

l6-575873 

199-43 

17.296578 

200.98 

18.023010 

202.6l 

18.755449 

204.32 

34 

16.587839 

199.45 

17.308637 

201.00 

18.035167 

202.64 

18.767709 

204.35 

35 

16.599807 

199.48 

17.320698 

2OI.O3 

18.047326 

202.66 

18.779971 

204.37 

36 

16.611776 

199.50 

17.332761 

201.06 

18.059487 

202.69 

18.792234 

204.40 

37 

16.623747 

199-53 

17.344825 

2OI.O8 

18.071649 

202.72 

18.804499 

204.43 

38 

16.635719 

J99-55 

17.356891 

201.  II 

18.083813 

202.75 

18.816767 

204.46 

39 

16.647693 

199.58 

17.368959 

201.14 

18.095979 

202.78 

18.829036 

204.49 

40 

16.659669 

199.60 

17.381028 

201.  l6 

18.108146 

202.80 

18.841305 

204.52 

41 

16.671646 

199.63 

17.393098 

201.19 

18.120315 

202.83 

18.853577 

204.55 

42 

16.683624 

199.65 

17.405171 

201.22 

18.132486 

202.86 

18.865851 

204.58 

43 

16.695604 

199.68 

17.417245 

201.24 

18.144658 

202.89 

18.878127 

204.61 

44 

16.707586 

199.70 

17.429320 

201.27 

18.156832 

202.92 

18.890404 

204.64 

45 

16.719569 

J99-73 

17.441397 

201.30 

18.169008 

202.94 

18.902684 

204.67 

46 

*6-73f553 

199.76 

17.453476 

201.32 

18.  181186 

202.97 

18.914965 

204.70 

47 

l6-743539 

199.78 

17.465556 

201-35 

18.193365 

203.00 

18.927247 

204.73 

48 

16.755527 

199.81 

17.477638 

201.38 

18.205546 

203.03 

18.939532 

204.76 

49 

16.767516 

199.83 

17.489722 

201.41 

18.217728 

203.06 

18.951818 

204.79 

50 

16.779507 

199.86 

17.501807 

201.43 

18.229912 

203.08 

18.964106 

204.81 

51 

16.791499 

199.88 

17.513894 

201.46 

18.242098 

203.11 

18.976396 

204.84 

52 

16.803493 

199.91 

17.525982 

201.49 

18.254286 

203.14 

18.988687 

204.87 

53 

16.815488 

199.94 

17.538072 

201.51 

18.266475 

203.17 

19.000981 

204.90 

54 

16.827485 

199.96 

17.550163 

201.54 

18.278666 

203.20 

19.01  3276 

204.93 

55 

16.839484 

199.99 

17.562257 

201.57 

18.290859 

203.23 

19.025573 

204.96 

56 

16.851484 

200.01 

17.574352 

201.59 

18.303053 

203.25 

19.037871 

204.90 

57 

16.863485 

200.04 

17.586448 

201.62 

18.315249 

203.28 

19.050172 

205.02 

58 

16.875488 

2OO.O6 

17.598546 

201.65 

18.327447 

203.31 

19.062474 

205.05 

59 

16.887493 

200.09 

17.610646 

201.68 

18.339646 

203.34 

19.074778 

205.08 

60 

16.899499 

200.12 

17.622747 

201.70 

18.351847 

203.37 

19.087084 

205.11 

572 


TABLE  VI, 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

28° 

29° 

30° 

31° 

M. 

Diff.  1". 

M. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0' 

19.087084 

205.11 

19.828747 

206.94 

1.313  3849 

44.08 

1.329  0430 

42.92 

I 

19.099391 

205.14 

19  841164 

206.97 

.313  6493 

44.06 

.329  3004 

42.91 

2 

19.111701 

205.17 

I9-853583 

207.00 

.313  9136 

44.04 

•329  5578 

42.89 

3 

19  124012 

205.20 

19.866004 

207.03 

.314  1778 

44.02 

.329  8151 

42.87 

4 

19.136325 

205.23 

19.878427 

207.06 

.314  4419 

44.00 

.330  0723 

42.85 

5 

19  148639 

205  26 

19  890852 

207.09 

1.314  7058 

43.98 

1.330  3293 

42.83 

6 

19.160956 

205.29 

19.903279 

207.13 

.314  9696 

43.96 

.330  5862 

42.81 

7 

19.173274 

205.32 

19.915707 

207.16 

•3X5  2333 

43-94 

•33°  8431 

42.80 

8 

19.185594 

205-35 

19.928137 

207.19 

.315  4969 

43.92 

.331  0998 

42.78   j 

9 

19.197916 

205.38 

19.940569 

207.22 

.315  7604 

43-90 

•331  3564 

42.76 

10 

19.210240 

205.41 

19.953003 

207.25 

1.316  0237 

43.88 

1.331  6129 

42.74 

11 

19  222566 

205.44 

19.965439 

207.28 

.316  2869 

43.86 

.331  8693 

42.72 

12 

19.234893 

205.47 

19.977877 

207.31 

.316  5500 

43-84 

•332  1255 

42.70 

13 

19.247222 

205.50 

19.990317 

207.34 

.316  8130 

43.82 

•332  3817 

42.69 

14 

!9-259553 

205-53 

20.002759 

207.38 

.317  0759 

43.80 

•332  6378 

42.67 

15 

19.271885 

205.56 

20.015202 

207.41 

1.317  3386 

43.78 

1.332  8937 

42.65 

16 

19.284220 

205.59 

20.027647 

207.44 

.317  6013 

43.76 

•333  H96 

42.63 

17 

18 

19.296556 
19.308894 

205.62 
205.65 

20.040095 
20.052544 

207.47 
207.50 

.317  8638 
.318  1262 

43-74 
43-72 

•333  4°53 
-333  6609 

42.61 
42.59 

19 

19.321234 

205.68 

20.064995 

207.53 

.318  3885 

43-70 

•333  9l64 

42.58 

20 

19.333576 

205.71 

20.077448 

207.57 

1.318  6506 

43.68 

1.334  1718 

42.56 

21 

19.345920 

205.74 

20.089903 

207.60 

.318  9127 

43.67 

•334  4271 

42.54 

22 

19.358265 

205.77 

20.102360 

207.63 

.319  1746 

43-65 

•334  6823 

42.52 

23 

19  370612 

205.80 

20.114818 

207.66 

.319  4364 

43.63 

•334  9374 

42.50 

24 

19.382961 

205.83 

20.127279 

207.69 

.319  6981 

43.61 

•335  1924 

42-49 

25 

19.395312 

205.86 

20.139741 

207.72 

I-3I9  9597 

43-59 

1-335  4472 

42.47 

20 

19.407665 

205.89 

20.152206 

207.76 

.320  2212 

43-57 

-335  7020 

42.45 

27 

19.420019 

205.92 

20.164672 

207.79 

.320  4825 

43-55 

•335  9567 

42.43 

28 

19432375 

205.95 

20.177140 

207.82 

.320  7438 

43-53 

.336  2112 

42.41 

29 

1  9-4447  34 

205.98 

20.189610 

207.85 

.321  0049 

4351 

.336  4656 

42.40 

30 

19.457094 

206.01 

20.202082 

207.88 

I.32I  2659 

43-49 

1.336  7199 

42.38 

31 

19.469455 

206.04 

20.214556 

207.91 

•  3*1  5268 

43-47 

.336  9742 

42.36 

32 

19.481819 

206.08 

20.227032 

207.95 

.321  7875 

43-45 

-337  2283 

42-34 

33 

19.494184 

206.  1  1 

20.239510 

207.98 

.322  0482 

43-43 

•337  4823 

42.33 

34 

19.506551 

206.14 

20.251989 

208.01 

.322  3087 

43-41 

•337  7362 

42.31 

35 

19.518921 

206.17 

20.264471 

208.04 

1.322  5692 

43.40 

1-337  99°° 

42.29 

36 

19.531292 

206.20 

20.276954 

208.07 

.322  8295 

43-38 

.338  2437 

42.27 

37 

19.543664 

206.23 

20.289440 

208.  II 

.323  0897 

43-36 

.338  4972 

42.25 

38 

19.556039 

206.26 

20.301927 

208.14 

•3*3  3498 

43-34 

•338  75°7 

42.24 

39 

19.568415 

206.29 

20.314416 

208.17 

.323  6097 

43.32 

•339  0041 

42.22 

40 

19.580794 

206.32 

20.326907 

208.20 

1.323  8696 

43-3° 

i-339  2573 

42.20 

41 

19.593174 

206.35 

20.339400 

208.24 

.324  1294 

43.28 

•339  5I05 

42.18 

42 

19.605556 

206.38 

20.351895 

208.27 

.324  3890 

43.26 

•339  7635 

42.17 

43 

19.617939 

206.41 

20.364392 

208.30 

.324  6485 

43.24 

.340  0165 

42.15 

44 

19.630325 

206.44 

20.376891 

208.33 

.324  9079 

43.22 

•340  2693 

42.13 

45 

19.642713 

206.47 

20.389392 

208.36 

1.325  1672 

43.21 

1.340  5221 

42.11 

46 

47 

19.655102 
19.667493 

206.50 
206.53 

20.401895 
20.414399 

208.39 
208.43 

.325  4263 
.325  6854 

43.19 
43-17 

.340  7747 
.341  0272 

42.10 
42.08 

48 

19.679886 

206.57 

20.426906 

208.46 

•325  9443 

43-15 

.341  2796 

42.06 

19 

19.692281 

206.60 

20.439415 

208.49 

.326  2032 

43-13 

-341  53'9 

42.04 

50 

19.704678 

206.63 

20.451925 

208.52 

1.326  4619 

43.11 

1.341  7841 

42.03 

51 

19.717076 

206.66 

20.464437 

208.56 

.326  7205 

43-09 

.342  0362 

42.01 

52 

19.729477 

206.69 

20.476952 

208.59 

.326  9790 

43-07 

.342  2882 

41.99 

53 

19.741879 

206.72 

20.489468 

208.62 

.327  2374 

43-05 

.342  5401 

41-97 

54 

19.754283 

206.75 

20.501986 

208.65 

•3*7  4957 

43.04 

.342  7919 

41.96 

55 

19.766689 

206.78 

20.514506 

208.69 

1.327  7538 

43.02 

1-343  °436 

41.94 

56 

19.779097 

206.81 

20.527029 

208.72 

.328  0119 

43-0° 

•343  2952 

41.92 

57 

19.791507 

206.84 

20.539553 

208.75 

.328  2698 

42.98 

•343  5467 

41.90 

58 

19.803919 

206.88 

20.552079 

208.78 

.328  5276 

42.96 

•343  798o 

41.89 

59 

19.816332 

206.91 

20.564607 

208.82 

.328  7853 

42.94 

•344  °493 

41.87 

60 

19.828747 

206.94 

20.577137 

208.85 

1.329  0430 

42.92 

1.344  3005 

41.85 

573 


TABLE  VI. 

b  or  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

32° 

33° 

34° 

35° 

logM. 

Diflf.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0' 

1.344  3005 

41.85 

1.359    1859 

40.86 

1-373  7*5i 

39-93 

1.387   9418 

39.06 

1     -344  55*5 

41.84 

•359  43*o 

40.84 

•373  9646 

39.91 

.388    1761 

39-°5 

2     .344  8025 

41.82 

•359  676o 

40.82 

-374  *°4i 

3990 

.388   4104 

39.04 

3     -345  0534 

41.80 

•359  9*°9 

40.81 

•374  4434 

39.88 

.388   6446 

39.02 

4 

•345  3°4i 

41.78 

.360  1657 

40.79 

•374  6827 

39.87 

.388    8787 

39.01 

5 

1.345  5548 

41-77 

1.360  4104 

40.78 

1.374  9218 

39-85 

1.389    1127 

38-99 

6 

•345  8053 

4*-75 

.360  6550 

40.76 

•375   l6°9 

39-84 

.389   3466 

38.98 

7 

.346  0558 

41-73 

.360  8995 

40.74 

•375  3999 

39.82 

.389   5804 

38.97 

8 

.346  3061 

41.72 

.361   1439 

4°-73 

.375  6388 

39.81 

.389   8x42 

38.95 

9 

.346  5564 

41.70 

.361   3883 

40.71 

.375  8776 

39-79 

.390  0479 

38.94 

10 

1.346  8065 

41.68 

1.361  6325 

40.70 

1.376  1164 

39-78 

1.390  2815 

3893 

11 

•347  0565 

41.66 

.361   8766 

40.68 

.376  3550 

39-77 

•39°  5*5° 

38.91 

12 

•347  3°65 

41.65 

.362  1207 

40.66 

•376  5935 

39-75 

.390  7484 

38.90 

13 

•347  5563 

41.63 

.362  3646 

40.65 

.376  8320 

39-74 

•39°  9817 

38.88 

14 

.347  8060 

41.61 

.362  6084 

40.63 

•377  0703 

39-7* 

.391  2150 

38.87 

15 

1.348  0557 

41.60 

1.362  8522 

40.62 

1.377  3086 

39-71 

1.391  4482 

38.86 

16 

•348  3°5* 

41.58 

.363  0959 

40.60 

•377  5468 

39-69 

.391  6813 

38.84 

17 

.348  5546 

41.56 

•363   3394 

40.59 

•377  7849 

39.68 

.391  9143 

38.83 

18 

.348  8040 

4!-55 

.363  5829 

40.57 

378  0230 

39.66 

.392  1472 

38.82 

19 

•349  °53* 

4'-53 

.363  8263 

40.56 

.378  2609 

39-65 

-39*  3801 

38.80 

20 

1.349  3023 

41.51 

1.364  0696 

40.54 

1.378  4987 

39.64 

1.392  6128 

38.79 

21 

•349  5513 

41.50 

.364  3128 

40.52 

.378  7365 

39.62 

•39*  8455 

38.77 

22 

•349  8o°3 

41.48 

•364  5559 

40.51 

.378  9742 

39.61 

•393  078i 

38.76 

23 

.350  0491 

41.46 

.364  7989 

40.49 

•379  *"7 

39-59 

•393  3I07 

38.75 

24 

.350  2978 

41.45 

.365  0418 

40.48 

•379  449* 

39.58 

•393  5431 

38.73 

25 

1.350  5464 

41.43 

1.365  2846 

40.46 

1.379  6866 

39-56 

1-393  7755 

38.72 

26 

•35°  7950 

41.41 

•365  5*73 

40.45 

•379  9*4° 

39-55 

•394  °°78 

38.71 

27 

.351  0434 

41.40 

.365  7699 

40.43 

.380  1612 

39-53 

•394  *4°° 

38.69 

28 

.351  2917 

41.38 

.366  0125 

40.41 

.380  3983 

39-5* 

•394  47*i 

38.68 

29 

•351  5399 

41.36 

.366  2549 

40.40 

.380  6354 

39.50 

•394  7041 

38.67 

30 

1.351  7880 

41-35 

1.366  4973 

40.38 

1.380  8724 

39-49 

1.394  9361 

38.65 

31 

.352  0361 

4'-33 

.366  7395 

4°-37 

.381   1093 

39-47 

.395   1680 

38.64 

1     32 

.352  2840 

41.31 

.366  9817 

4°-35 

.381   3461 

39-46 

•395  3998 

38-63 

I     33 

•35*  53'8 

41.30 

.367  2238 

40.34 

.381   5828 

39-45 

•395  63'5 

38.61 

34 

•35*  7795 

41.28 

.367  4657 

40.32 

.381   8194 

39-43 

-395  8631 

38.60 

35 

1.353  o*7* 

41.26 

1.367  7076 

40.31 

1.382  0559 

39-4* 

1.396  0947 

38.59 

36 

•353  *747 

41.25 

•367  9494 

40.29 

.382  2924 

39.40 

.396  3262 

38-57 

37 

•353  5**i 

41.23 

.368   1911 

40  28 

.382  5288 

39-39 

.396  5576 

38.56 

38 

•353  7694 

41.21 

.368  4327 

40.26 

.382  7651 

39-37 

.396  7889 

38.55 

39 

•354  Ol67 

41.20 

.368  6742 

40.25 

.383  0013 

39-36 

.397    0201 

38.53 

40 

1.354  2638 

41.18 

1.368  9157 

40.23 

1.383  2374 

39-35 

1.397    2513 

38.52 

41 

•354  5IQ8 

41.16 

.369  1570 

40.21 

•383  4734 

39-33 

•397  4823 

38-51 

42 

•354  7578 

41.15 

•369  3983 

40.20 

•383   7°93 

39.32 

•397  7133 

38.49 

43 

•355  °°46 

41-13 

•369  6394 

40.18 

•383  94-5* 

39-3° 

•397  944* 

38.48 

44 

•355  *5'3 

41.11 

.369  8805 

40.17 

.384  1809 

39-*9 

.398   1751 

38-47 

45 

1.355  498° 

41.10 

1.370  1214 

40.15 

1.384  4166 

39.27 

1.398  4058 

38.45 

46 

•355  7445 

41.08 

.370  3623 

40.14 

.384  6522 

39.26 

•398  6365 

38.44 

47 

•355  99°9 

41.07 

.370  6031 

40.12 

.384  8878 

39-*5 

.398  8671 

38-43 

48 

.356  2373 

41.05 

.370  8438 

40.11 

.385   1232 

39*3 

-399  0976 

38.41 

49 

.356  4836 

41.03 

.371  0844 

40.09 

•385  3585 

39.22 

-399  3*8i 

38.40 

50 

1.356  7297 

41.02 

1.371  3249 

40.08 

1.385  5938 

39.20 

1.399  5584 

38-39 

51 

•356  9758 

41.00 

•371  5654 

40.06 

.385  8290 

39-19 

•399  7887 

38.37 

52 

•357  **'7 

40.98 

.371  8057 

40.05 

.386  0641 

39.18 

.400  0189 

38.36 

53 

•357  4676 

40.97 

•37*  0459 

40.03 

.386  2991 

39.16 

.400  2491 

38.35 

54 

•357  7134 

40.95 

.372  2861 

40.02 

.386  5340 

39-15 

.400  4791 

38.33 

55 

i-357  959° 

40.94 

1.372  5261 

40.00 

1.386  7689 

39-13 

1.400  7091 

38.32 

56 

.358  2046 

40.92 

.372  7661 

39-99 

.387  0036 

39.12 

.400  9390 

38-31 

57 

.358  4501 

40.90 

.373  0060 

39-97 

-387  *383 

39-" 

.401   1688 

.38.30 

58 

.358  6954 

40.89 

•373  *458 

39.96 

•387  47^9 

39.09 

.401   3985 

38.28 

59 

.358  9407 

40.87 

•373  4855 

39-94 

.387  7074 

39.08 

.401  6282 

38.27 

60 

1.359  1859 

40.86 

1.373  7251 

39-93 

1.387  9418 

39.06 

1.401   8578 

38.26 

574 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

36° 

37° 

38° 

39°    | 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

O' 

1.401  8578 

38.26 

1.415  4930 

37-5° 

1.428  8662 

36.80 

1.441  9943 

36.14 

1 

.402  0873 

38.24 

.415  7180 

37-49 

.429  0869 

36-79 

.442  21  1  I 

36.13 

2 

.402  3167 

38-23 

.415  9429 

37-47 

.429  3076 

36.78 

.442  4.279 

36.12 

3 

.402  5460 

38.22 

.416  1678 

37-46 

.429  5281 

36-77 

.442  6446 

36.11 

4 

.402  7753 

38.20 

.416  3925 

37-45 

.429  7488 

36-75 

.442  86l2 

36.10 

5 

1.403  0045 

38.19 

1.416  6172 

37-44 

1.429  9693 

36.74 

1-443  0778 

36  co 

6 

.403  2336 

38.18 

.416  8419 

37-43 

.430  1897 

36.73 

•443  2943 

36.08 

7 

.403  4626 

38.17 

.417  0664 

37-41 

.430  4101 

36.72 

.443  5107 

36.07 

8 

.403  6916 

38.15 

.417  2909 

37.40 

•43°  6304 

36-71 

•443  7271 

36.06 

9 

.403  9205 

38.14 

•417  5153 

37-39 

.430  8506 

36-7° 

•443  9434 

36.05 

10 

1.404  1493 

38.13 

1-4*7  7396 

37-38 

1.431  0708 

36.69 

1.444  1597 

36-04 

11 

.404  3780 

38.12 

•4i7  963.9 

37-37 

•431  2909 

36.68 

•444  3758 

36-03 

12 

.404  6067 

38.10 

.418  1881 

37-36 

.431  5109 

36.66 

•444  5920 

36.02 

13 

.404  8352 

38.09 

.418  4122 

37-35 

.431  7308 

36.65 

.444  8080 

36.00 

14 

.405  0637 

38.08 

.418  6362 

37-33 

•431  95°7 

36.64 

.445  0240 

35-99 

15 

1.405  2921 

38.06 

1.418  8602 

37.32 

1.432  1705 

36.63 

1.445  2400 

35-98 

16 
17 

.405  5205 
.405  7488 

38.05 
38.03 

.419  0841 
.419  3079 

37-31 
37.3° 

-432  39°3 
.432  6100 

36.62 
36.61 

•445  4558 
•445  6716 

35-97 
35-96 

18 

.405  9769 

38.02 

•419  5317 

37-29 

.432  8296 

36.60 

.445  8874 

35-95 

19 

.406  205  i 

38.01 

.419  7554 

37-27 

.433  0491 

36.59 

.446  1031 

35-94 

20 

1.406  4331 

38.00 

1.419  9790 

37.26 

1.433  2686 

36.57 

1.446  3187 

3593 

21 

.406  66  1  1 

37-99 

.420  2026 

37-25 

.433  4881 

36.56 

•446  5343 

3592 

22 

.406  8889 

37-97 

.420  4260 

37-24 

-433  7°74 

36-55 

.446  7498 

35-91 

23 

.407  1  1  68 

37.96 

.420  6494 

37-23 

-433  9267 

36.54 

.446  9652 

35-90 

24 

•407  3445 

37-95 

.420  8728 

37.22 

.434  1459 

36.53 

.447  1806 

35-89 

25 

1.407  5721 

37-94 

1.421  0960 

37-20 

1.434  3651 

36-52 

1-447  3959 

35-88 

26 

.407  7997 

37-92 

.421  3192 

37-19 

•434  5842 

36-51 

.447  6112 

35.87 

27 

.408  0272 

37-91 

.421  5423 

37-i8 

•434  8032 

36.50 

.447  8263 

35.86 

28 

.408  2547 

37.90 

.421  7654 

37.17 

.435  0221 

36.49 

.448  0415 

35.85 

29 

.408  4820 

37.89 

.421  9884 

37.16 

•435  2410 

36.48 

.448  2565 

35-84 

30 

1.408  7093 

37.87 

1.4*2  2113 

37.15 

1-435  4598 

36.47 

1.448  4715 

35-83 

31 
32 
33 

.408  9365 
.409  1636 
.409  3907 

37-86 

37.85 
37.84 

.422  A34i 
.422  6569 
.422  8796 

37-13 
37-12 
37-11 

.435  6786 
•435  8973 
•436  H59 

36-46 
36.44 

36.43 

.448  6865 
.448  9014 
.449  1162 

35-82 
35.81 
35.80 

34 

.409  6177 

37.82 

.423  1  022 

37.10 

•436  3345 

36-42 

•449  33°9 

35-79 

35 

1.409  8446 

37.81 

1.423  3248 

37-09 

1.436  5530 

36-41 

1-449  5456 

35-78 

36 

.410  0714 

37.80 

•423  5473 

37.08 

.436  7714 

36.40 

•449  7603 

35-77 

37 

.410  2981 

37.78 

•423  7697 

37.06 

•436  9898 

36.39 

•449  9749 

35-76 

38 

.410  5248 

37-77 

.423  9920 

37-05 

.437  2081 

36.38 

.450  1894 

35-75 

39 

.410  7514 

37.76 

.424  2143 

37.04 

•437  4263 

36.37 

.450  4038 

35-74 

40 

1.410  9780 

37-75 

1.424  4365 

37-03 

i-437  6445 

36.36 

1.450  6182 

35-73 

41 

.411  2044 

37-74 

.424  6586 

37.02 

.437  8626 

36.35 

.450  8325 

35-72 

42 
43 

.411  4308 
.411  6571 

37-72 
37-71 

.424  8807 
.425  1027 

37.01 
36.99 

.438  0806 
.438  2986 

36-34 
36.32 

.451  0468 
.451  2610 

35-71 
35-70 

44 

.411  8833 

37-7° 

.425  3246 

36-98 

.438  5165 

36.31 

.451  4752 

35-69 

45 

1.412  1095 

37-69 

1.425  5465 

36.97 

1.438  7344 

36-30 

1.451  6893 

35.68 

<  46 

.412  3356 

37-68 

.425  7683 

36.96 

.438  9522 

36.29 

•451  9°33 

35.67 

47 

t   s- 

.412  5616 

37.66 

.425  9900 

36.95 

•439  1699 

36.28 

.452  1173 

35-66 

48 

.412  7875 

37-65 

.426  2117 

36-94 

•439  3875 

36.27 

.452  3312 

35-65 

49 

.413  0134 

37-64 

.426  4333 

36-92 

.439  6051 

36.26 

•452  5450 

35-64 

50 

1.413  2392 

37-63 

1.426  6548 

36-91 

1.439  8226 

36.25 

1.452  7588 

35-63 

51 

.413  4649 

37.61 

.426  8762 

36.90 

.440  0401 

36-24 

.452  9725 

35-62 

52 
53 
54 

.413  6905 
.413  9161 
.414  1416 

37.60 
37-59 
37.58 

.427  0976 

•427  3l89 
.427  5402 

36.89 
36.88 
36.87 

.440  2575 
.440  4748 
.440  6921 

36.23 
36.22 
36.20 

.453  1862 
•453  3998 
•453  6134 

35.60 
35-59 

55 
56 

1.414  3670 

37.56 
37-55 

1.427  7613 
.427  9824 

36.86 
36-85 

1.440  9093 
.441  1264 

36-19 
36.18 

1.453-  8269 
.454  0403 

35-57 

57 

.414  8176 

37.54 

.428  2035 

36-83 

•441  3436 

36.17 

•454  2537 

35-56 

58 
59 

.415  0429 
.415  2680 

37-53 
37-51 

.428  4244 
.428  6453 

36.82 
36.81 

.441  5605 
•441  7774 

36.16 
36.I5 

•454  4670 
.454  6802 

35-55 
35-54 

60 

1.415  4930 

37-5° 

1.428  8662 

36.80 

1.441  9943 

36.14 

1-454  *934 

35-53 

075 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

40° 

41° 

42° 

43° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

O' 

1.454   8934 

35-53 

1.467   5782 

34-95 

1.480  0627 

34-41 

T-492  3597 

33-91 

1 

•455   1065 

35-51 

.467   7879 

34-94 

.480  2691 

34.40 

.492  5631 

33-90 

2 

•455  3*96 

35-51 

.467   9976 

34-93 

.480  4755 

34.40 

.492  7665 

3389 

3 

.455  5326 

35-5° 

.468   2071 

34.92 

.480  6819 

34-39 

.492  9698 

33-88 

4 

•455  7456 

35-49 

.468   4166 

34-  9  l 

.480  8882 

34-38 

•493   i73i 

33-87 

5 

T-455  9585 

35.48 

1.468   6261 

34.90 

1.481   0944 

34-37 

r-493   3764 

33-87 

6 

.456  1713 

35-47 

.468    8355 

34-9° 

.481   3006 

34-36 

•493  5796 

33.86      | 

7 

.456  3841 

35-46 

•469   °448 

34.89 

.481   5068 

34-35 

•493  7827 

33-85 

8 

.456  5968 

35-45 

.469    2541 

34.88 

.481   7129 

34-34 

•493  9858 

33-84      1 

9 

.456  8094 

35-44 

.469   4634 

34.87 

.481   9189 

34-33 

.494  1888 

33-83 

10 

1-457    0220 

35-43 

1.469    6725 

34.86 

1.482  1249 

34-33 

1.494  39l8 

33.83 

11 

•457  2346 

35-42 

.469    8817 

34-85 

.482  3308 

34-32 

•494  5948 

33-82 

12 

•457  4470 

35-41 

.470  0907 

34.84 

.482  5367 

34-31 

•494  7977 

33-8i 

13 

•457  6595 

35-40 

.470   2998 

34.83 

.482  7425 

34-3° 

.495  0005 

33.80 

14 

•457  8718 

35-39 

.470   5087 

34.82 

.482  9483 

34-29 

•495  2°33 

33-79 

15 

1.458  0841 

35.38 

1.470  7176 

34.81 

1.483  1540 

34.28 

1.495  4°6i 

33-79 

16 

.458  2964 

35-37 

.470   9265 

34.80 

•483  3597 

34.28 

.495  6088 

33-78 

17 

.458  5086 

35-36 

•471    1353 

34-79 

•483   5653 

34-27 

-495  8114 

33-77 

18 

.458  7207 

35-35 

.471    3440 

34-79 

•483  77°9 

34.26 

.496  0140 

33.76 

19 

.458  9328 

35-34 

•471  55*7 

34.78 

.483  9764 

34-25 

.496  2166 

33-75 

20 

1.459  H48 

35-33 

1.471  7613 

34-77 

1.484  1819 

34-24 

1.496  4191 

33-75 

21 

•459  3567 

35.32 

.471  9699 

34.76 

.484  3873 

34-23 

.496  6216 

33-74 

22 

.459  5686 

35-31 

.472  1784 

34-75 

.484  5927 

34-22 

.496  8240 

33-73 

23 

•459  7805 

35-3° 

•472  3869 

34-74 

.484  7980 

34-22 

.497  0264 

33-72 

24 

•459  99" 

35-^9 

•472  5953 

34-73 

.485  0033 

34.21 

•497  2287 

33.71 

25 

1.460  2040 

35.28 

1.472  8037 

34-73 

1.485  2085 

34.20 

1.497  4310 

33-71 

26 

.460  4156 

35.27 

.473    0120 

34-72 

•485  4'37 

34-19 

•497  6332 

33.70 

27 

.460  6272 

35.26 

.473    2203 

34.71 

.485  6188 

34.18 

•497  8354 

33-69 

28 

.460  8388 

35-25 

•473    4285 

34-7° 

•485  8239 

34-17 

.498  0376 

33-68 

29 

.461  0503 

35-24 

•473  6366 

34.69 

.486  0289 

34.16 

.498  2396 

33.68 

30 

1.461  2617 

35-23 

1.473  8447 

34.68 

1.486  2338 

34.16 

1.498  4417 

33.67 

31 

.461  4731 

35-23 

•474  °527 

34-67 

.486  4388 

3415 

.498  6437 

33.66 

32 

.461  6844 

35-22 

.474  2607 

34-66 

.486  6436 

34-14 

.498  8456 

33.65 

33 

.461  8957 

35-21 

.474  4686 

34.65 

.486  8484 

34.13 

•499  °475 

33-65 

34 

.462  1069 

35.20 

•474  6765 

34.64 

•487  <>532 

34.12 

.499  2494 

33-64 

35 

1.462  3180 

35-J9 

1.474  8843 

34-63 

1.487  2579 

34.12 

1.499  4512 

33-63 

36 

.462  5291 

35-i8 

.475  0921 

34.62 

.487  4626 

34-  " 

•499  653o 

33.62 

37 

.462  7401 

35-J7 

•475  2998 

34.61 

.487  6672 

34.10 

•499  8547 

33.62 

38 

.462  9511 

35-i6 

-475   5°75 

34.61 

.487  8718 

34.09 

.500  0563 

33.61 

39 

.463  1620 

35-15 

•475  7J5i 

34.60 

.488  0763 

34.08 

.500  2580 

33.60 

40 

1.463  3729 

35-H 

1.475  9227 

34-59 

1.488  2807 

34.07 

1.500  4595 

33-59 

41 

•463   5837 

35-x3 

.476   1302 

34.58 

.488  4852 

34.07 

.500  6611 

33.58 

42 

•463  7944 

35-12 

.476  3376 

34-57 

.488  6895 

34.06 

.500  8625 

33-58 

43 

.464  0051 

35-11 

.476  5450 

34-56 

.488  8939 

34.05 

.501  0640 

33-57 

44 

.464  2158 

35-i° 

.476  7524 

34-55 

.489  0981 

34-°4 

.501   2654 

33-56 

45 

1.464  4263 

35-°9 

1.476  9596 

34-54 

1.489  3023 

34-03 

1.501  4667 

33-55 

46 

.464  6369 

35.08 

•477  1669 

34-54 

•489  5°65 

34.02 

.501   6680 

33-55 

47 

.464  8473 

35.07 

•477  374i 

34-53 

.489  7106 

34.02 

.501   8693 

33-54 

48 

•465  °577 

35.06 

•477  5812 

34-5* 

.489  9147 

34.01 

.502  0705 

33-53 

49 

.465  2681 

35-°5 

•477  7883 

34-5  i 

.490  1187 

34.00 

.502  2716 

33-5a 

50 

1.465  4784 

35-°4 

1-477  9953 

34-5° 

1.490  3227 

33-99 

1.502  4727 

33-51 

51 

.465   6886 

35-°4 

.478  2023 

34-49 

.490  5266 

33.98 

.502  6738 

33-51 

52 

.465   8988 

35-°3 

.478  4092 

34-48 

•49°  73°5 

33-97 

.502  8748 

33-5° 

53 

.466   1090 

35.02 

.478  6161 

34-47 

.490  9343 

3396 

•5°3  0758 

33-49 

54 

.466  3190 

35>01 

.478  8229 

34.46 

.491   1381 

33-95 

.503  2767 

33-48 

55 

1.466   5290 

35.00 

1.479  0297 

34-46 

1.491   3418 

33-95 

1.503  4776 

33-48 

56 

.466  7390 

34-99 

•479  2364 

34-45 

•491   5455 

33-94 

.503  6784 

33-47 

57 

.466  9489 

34.98 

.479  4430 

34-44 

•49  *   749  * 

33-93 

•5°3  8792 

33-46 

58 

.467   1587 

34-97 

•479  6496 

34-43 

.491   9527 

3392 

.504  0800 

33-45 

59 

.467   3685 

34.96 

.479  8562 

34.42 

.492   1562 

33.91 

.504  2807 

33-44 

60 

1.467  5782 

34-95 

1.480  0627 

34.41 

!-49a  3597 

33.91 

1.504  4813 

33-44 

576 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Oibit. 


V. 

44° 

45° 

4:6° 

47° 

log  M. 

Diff.  1". 

logM. 

Diff.  I". 

logM. 

Diff.  I". 

logM. 

Diff.  1". 

0' 

1.504  4813 

33-44 

1.516   4390 

33.00 

1.528   2435 

3*59 

*-539  9°48 

32.20 

1 

.504   6819 

33-43 

.516   6370 

32.99 

.528   4390 

32.58 

.540  0980 

32.20 

2 

.504   8825 

33  4* 

.516   8349 

32.98 

.528   6344 

32.57 

.540  2912 

32.19 

3 

.505    0830 

33-4^ 

.517   0328 

3298 

.528   8299 

32.57 

.540  4843 

32.18 

4 

.505    2835 

33-41 

.517   2306 

32.97 

.529  0252 

32.56 

.540  6774 

32.18 

5 
6 

1.505   4839 
.505    6843 

3340 
33-39 

1.517   4284 
.517   6262 

32.96 
32.96 

1.529   2206 
.529  4159 

3M5 

3255 

1.540  8705 
.541  0635 

32-I7 
32.17 

7 

.505    8846 

33-39 

.517    8239 

32-95 

.529   6112 

32-54 

.541  2564 

32.16 

8 

.506   0849 

33-3^ 

.518   0216 

3*94 

.529   8064 

32.53 

.541  4494 

32.15 

9 

.506   2852 

33  37 

.518   2192 

32-93 

.530  0016 

32-53 

.541   6423 

32.15 

10 
11 

1.506   4854 
.506   6855 

33-36 
33-36 

1.518   4168 
.518   6143 

32.93 
3292 

1.530   1967 
.530  3918 

32.52 

32-51 

1.541   8352 
.542  0280 

32.14 
32.14 

12 

.506    8856 

33-35 

.518  8118 

32.91 

.530   5869 

32-51 

.542  2208 

32.13 

13 

.507   0857 

33-34 

.519  0093 

32.91 

•53°  78*9 

32.50 

•542  4135 

32.12 

14 

.507   2857 

33-33 

.519  1067 

32.90 

.530  9769 

32.49 

.542  6063 

32.11 

15 

1.507   4857 

33-33 

1.519  4041 

32.89 

1.531    1719 

32.49 

1.542  7989 

32.11 

16 

.507   6856 

33-32 

.519  6014 

32.89 

.531   3668 

32.48 

•542  99l6 

32.10 

17 

.507   8855 

33-31 

.519  7987 

32.88 

.531   5616 

32.48 

•543   l842 

32.10 

18 

.508   0853 

33-3° 

.519  9960 

32-87 

•531  7565 

32.47 

•543  3768 

32.09 

19 

.508    2851 

33.29 

.520  1932 

32.86 

•531  95J3 

32.46 

•543  5693 

32.09 

20 

1.508   4849 

13.29 

1.520  3904 

3286 

1.532  1460 

32.46 

1.543  76i8 

32.08 

21 

.508    6846 

33.28 

.520  5875 

32.85 

.532  3407 

32-45 

•543  9543 

32-08 

22 

.508    8843 

33-27 

.520  7846 

32.84 

•532  5354 

32-44 

•544  1467 

32.07 

23 

.509   0839 

33-27 

.54.0  9816 

32.84 

.532  7300 

3244 

•544  339i 

32.06 

24 

.509   2835 

33.26 

.521    1786 

32.83 

.532  9246 

32-43 

•544  53*5 

32.06 

25 

1.509  4830 

33-25 

1.521   3756 

32.82 

1.533   "92 

32.43 

1.544  7238 

32.05 

26 

.509   6825 

33-14 

.521   5725 

32.82 

•533  3J37 

32-42 

•544  9*6i 

32.04 

27 

.509   8819 

33  24 

.521   7694 

32.81 

•533  5°*2 

32.42 

•545   '°83 

32.04 

28 

.510  0813 

33,23 

.521  9662 

32.80 

•533  7027 

32.41 

•545  3°°5 

32.03 

29 

.510  2807 

33.22 

.522  1630 

32.80 

•533  897i 

32.40 

•545  4927 

32.03 

30 

1.510  4800 

33,21 

1.522  3598 

32-79 

1.534  0914 

32.39 

1.545  6849 

32.02 

31 

.510   6792 

33-21 

.522  5565 

32.78 

.534  2858 

32-39 

.545  8770 

32.02 

32 

.510   8785 

33,20 

.522  7531 

32.78 

•534  4801 

32.38 

.546  0690 

32.01 

33 

.511    0776 

33-*9 

.522  9498 

32.78 

•534  6743 

32-37 

.546  2611 

32.00 

34 

.511    2768 

i3.i8 

.523   1464 

3*-77 

.534  8685 

32.37 

•546  4531 

32.00 

35 

1.511  4759 

33.18 

1.523  3429 

32.76 

1.535  0627 

32.36 

1.546  6450 

3*-99 

36 

.511   6749 

33-*7 

•523  5394 

32.75 

•535  2568 

32-35 

.546  8370 

31.98 

37 

.511   8739 

33.16 

•5*3  7359 

32.74 

•555  45°9 

32-35 

.547  0289 

31.98 

38 

.512  0729 

33-I5 

•5*3  93*3 

32-73 

•535  6450 

32-34 

•547  2207 

31.97 

39 

.512  2718 

33-15 

.524  1287 

32.73 

•535  839° 

32.33 

•547  4"5 

3J»97 

40 

1.512  4707 

33-H 

1.524  3251 

32.72 

1.536  0330 

32.33 

1.547  6043 

31.96 

41 

.512  6695 

33-!3 

.524  5214 

32.71 

.536  2270 

32.32 

.547  7961 

31.96 

42 

.512  8683 

33-n 

.524  7176 

32.71 

.536  4209 

32.32 

•547  9878 

31-95 

43 

.515  0670 

33-12 

.524  9138 

32.70 

.536  6148 

32.31 

.548   1795 

3i-94 

44 

.513  2657 

33-" 

.525   noo 

32.70 

.536  8086 

32.30 

.548   3711 

3'-94 

45 

1.513  4644 

33.11 

1.525   3062 

32.69 

1.537  0024 

32.30 

1.548  5627 

31-93 

46 

s  r 

.513  6630 

33.10 

.525   5023 

32.68 

•537   i962 

32.29 

•548  7543 

3*-93 

47 

.513  8615 

33.09 

.525  6983 

32.67 

-537  3899 

32.28 

.548  9458 

31.92 

48 

.514  0601 

3308 

.525   8944 

32.67 

•537  5836 

32.28 

•549   H73 

3*-9i 

49 

.514  2586 

33-°7 

.526  0903 

3266 

•537  7772 

32.27 

•549  S288 

3l'9l 

50 
51 

1.514  4570 
.514  6554 

33.07 
33.06 

1.526  2863 
.526  4822 

32.65 
32.64 

1.537  9708 
.538   1644 

32.26 
32.26 

1.549  5202 
•549  7"6 

31.90 

31.90 

52 

.514  8537 

33-°5 

.526  6780 

32.64 

•538   3579 

32.25 

•549  9°3° 

31.89 

53 

.515  0520 

33-°5 

.526  8739 

32.63 

•538  55H 

32.25 

.550  0943 

31.88 

rt  r> 

54 

•5*5  2503 

33-°4 

.527  0696 

32.62 

•538  7449 

32.24 

.550  2856 

31.88 

55 

1.515  4485 

33-°4 

1.527  2654 

32.62 

i-538  9383 

32.23 

1.550  4769 

31.87 

56 
57 

.515  6467 
.515  8449 

33-°3 
33.02 

.527  4611 
.527  6567 

32.61 
32.61 

•539  i3'7 
•539   325° 

32.23 
32.22 

.550  6681 
.550  8593 

31.87 
31.86 

58 

.516  0430 

33  oi 

.527  8524 

32.60 

•539  5l83 

32.21 

.551    050A 

31.86 

59 

.516  2410 

33.01 

.528  0479 

32.60 

•539  7"6 

32.21 

.551  2416 

3i-85 

60 

1.516  4390 

33.00 

1.528  2435 

32-59 

1.539  9048 

32.20 

1.551  4326 

3I-85 

577 


TABLE  VI, 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

48° 

49° 

50° 

51° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  l". 

0' 

1 

1.551   4326 
.551    6237 

31.85 
31.84 

1.562  8360 
.563   0250 

HI! 

1.574   1234 
•574  3T°6 

31.20 
31.20 

.585   4886 

30.91 
30.91 

2      .55i   8i47 

3I-83 

.563   2140 

3I-5° 

•574  4977 

3I-I9 

•585    6740 

30.90 

3     .552  0057 

.563   4030 

31.50 

•574  6849 

3I-I9 

.585    8594 

30.90 

4 

.552    1966 

31.82 

.563   5920 

3M9 

.574  8720 

31.18 

.586   0448 

30.89 

5 

1.552    3876 

31.82 

1.563   7809 

31.48 

1.575  0590 

31.18 

1.586   2302 

30.89 

6 

.552    5784 

31.81 

.563   9698 

31.48 

•575  2461 

31.17 

.586   4155 

30.89 

7 

.552   7693 

31.80 

.564    1586 

3  '-47 

•575  4331 

31.17 

.586   6008 

30.88 

9 

.552   9601 
•553   '508 

31.80 

•564  3475 
•564  5363 

3M7 
31.46 

.575  6201 
•575  8070 

31.16 
31.16 

.586   7859 
.586   9713 

30.87 
30.87 

10 

1-553  34l6 

31.79 

1.564  7250 

31.46 

J-575  9939 

31.15 

1.587    1565 

30-87      ; 

11 

•553  5323 

3I-78 

.564  9138 

3»«45 

.576   1808 

31.15 

•587    3417 

30.86 

12 

•553  723° 

31.78 

.565   1025 

3»-45 

.576   3677 

3I-I4 

.587    5268 

30.86 

13 

•553  9*36 

3'-77 

.565  2911 

3i-44 

.576  5546 

3I-I4 

.587   7120 

30.85 

14 

.554  1042 

31.76 

.565  4798 

3i-44 

.576  7414 

3I-13 

.587   8971 

30.85 

15 

1.554  2948 

31.76 

1.565  6684 

3i-43 

1.576  9281 

31.13 

1.588   0821 

30.84 

16 

•554  4853 

.565  8569 

•577   H49 

31.12 

.588   2672 

30.84 

17 

•554  675» 

31-75 

.566  0455 

31.42 

-577   3°l6 

31.12 

.588   4522 

30-83 

18 

•554  8663 

3I-74 

.566  2340 

31.41 

•577  4883 

31.11 

.588    6372 

30.83 

19 

•555  °567 

3'-74    ' 

.566  4225 

3J-41   ' 

•577  6749 

31.11 

.588    8222 

20 

1.555  2472 

3J-73     •• 

1.566  6109 

31.40 

1.577  8615 

31.10 

1.589   0071 

30.82 

21 

•555  4375 

.566  7993 

31.40 

.578  0481 

31.10 

.589    1920 

30.82 

22 

•555  6279 

31.72    - 

.566  9877 

31.39 

•578  2347 

31.09 

.589    3769 

30.81 

23 

•555  8182 

31.71     • 

.567   1761 

3r-39 

.578  4213 

31.09 

.589   5618 

30.81 

24 

.556  0084 

31.71      : 

•567   3644 

31.38 

.578  6078 

31.08 

.589   7466 

30.80 

25 

1.556  1987 

31.70     i 

1.567  5527 

31.38 

1.578  7942 

31.08 

1.589   9314 

30.80 

26 

.556  3888 

31.70 

.567  7409 

3i-37 

•578  9807 

31.07 

.590   1162 

30.79 

27 

•556  579° 

31.69 

.567  9291 

3*-37 

.579   1671 

31.07 

.590   3009 

30-79 

28 

.556  7691 

31.68 

.568   1173 

31.36 

•579  3535 

31.06 

•59°  4857 

30.78 

29 

.556  9592 

31.68 

.568  3055 

31.36    i 

•579  5399 

31.06 

•59°  67°4 

30.78 

30 

i-557  H93 

31.67      i 

1.568  4936 

31-35     •• 

1.579  7262 

31.06 

1.590  8550 

30.78 

31 

•557  3393 

31.67 

.568  6817 

•579  9I25 

3I>O5 

.591  0397 

3°-77 

32 

•557  5293 

31.66 

.568  8698 

3'-34 

.580  0988 

31.04 

.591  2243 

30-77 

33 

•557  7i93 

31.66 

.569  0579 

3>-34 

.580  2851 

31.04 

.591  4089 

30.76 

34 

•557  9092 

3I-65 

.569  2459 

31-33 

.580  4713 

3J-°3 

,59i  5935 

30.76 

35 

1.558  0991 

3*-65 

1.569  4338 

3J-33 

1.580  6575 

31.03 

1.591  7780 

30-75 

36 

.558  2890 

31.64 

.569  6218 

31.32    i 

.580  8436 

.591  9625 

30-75 

37 

•558  4788 

31.64 

.569  8097 

31.32    , 

.581  0298 

31.02 

.592  1470 

30-75 

38 

.558  6686 

31.63 

.569  9976 

.581  2159 

31.02 

•592  33*5 

30-74 

39 

•558  8584 

31.62 

.570  1854 

31.30    ( 

.581  4020 

31.01 

•592  5'59 

30.74 

40 

1.559  0482 

31.62 

i-570  3733 

31.30 

1.581  5.8*0 

31.01 

1.592  7003 

30-73 

41 

•559  2379 

31.61 

.570  5611 

31.29 

.581   7740 

31.00 

.592  8847 

3°-73 

42 

•559  4275 

31.61 

.570  7488 

31.29 

.581   9600 

31.00 

.593  0690 

30.72 

43 

•559  6172 

31.60 

.570  9366 

31.28 

.582   1460 

3°-99 

•593  2534 

30.72 

44 

.559  8068 

31.60 

.571   1243 

31.28 

.582  3319 

30.99 

•593  4377 

30-72 

45 

r-559  9963 

3'«59 

1.571  3119 

31.28 

1.582  5179 

30.98 

1.593  6219 

30.71 

46 

.560  1859 

3'-59 

.571  4996 

31.27 

-582  7037 

3098 

.593  8062 

30.71 

47 

•56°  3754 

31.58 

.571  6872 

31.27 

.582  8896 

3°-97 

•593  99°4 

30.70 

48 
49 

.560  5648 
.560  7543 

$*-57 

3i-57 

.571  8748 
.572  0623 

31.26 
31.26 

-583  °754 
.583  2612 

30.97 
30.96 

•594  1746 
•594  3588 

30.70 
30.69 

50 

1.560  9437 

3J-56 

1.572  2499 

31.25 

1.583  4470 

30.96 

1.594  5429 

30.69 

51 

.561    1331 

31.56 

•572  4373 

31.25 

.583  6327 

3°-95 

•594  7270 

30.68 

52 

.561  3224 

3J-55 

.572  6248 

31.24 

.583  8184 

3°-95 

•594  9111 

30.68 

53 

.561  5117 

3«»55 

.572  8123 

31.24 

.584  0041 

3°-94 

•595  0952 

30.68 

54 

.561  7010 

3'-54 

•572  9997 

31.23 

.584  1898 

30.94 

•595  2792 

30.67 

55 

1.561   8902 

3*-54 

1.573   1870 

31.23 

1-584  3754 

30.94 

i-595  4633 

30.67 

56 
57 

58 

.56*  0794 
.562  2686 
.562  4578 

si-si 

J»-53 

•573  3743 
•573  56»6 

•573  7489 

31.22 
31.22 
31.21 

.584  5610 
.584  7466 
•584  9321 

3°-93 
30-93 
30.92 

•595  6473 
•595  8312 
.596  0151 

30  66 
30.66 
30.65 

59 

.562  6469 

S'-S* 

•573  936z 

31.21 

.585   1176 

30.92 

>$g6   1990 

30.65 

60 

1.562  8360 

M-S- 

1.574  1234 

31.20 

i-585  3031 

30.91 

1.596  3829 

30.65 

578 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

52° 

53° 

54° 

55° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  I". 

logM. 

Diff.  1". 

O' 

1.596  3829 

30.65 

1.607  3703 

30.40 

1.618  2724 

30.17 

1.629  °959 

29.96 

1 

.596  5668 

30.64 

.607  5527 

30-39 

.618  4534 

30.17 

.629  2757 

29.96 

2 

.596  7506 

30.64 

.607  7350 

3°-39 

.618  6344 

30.16 

.629  4554 

29.96 

3 

•596  9344 

30.63 

.607  9174 

30-39 

.618  8153 

30.16 

.629  6351 

29.95 

4 

.597  1182 

30.63 

.608  0997 

30.38 

.618  9963 

30.16 

.629  8148 

29.95 

5 

1.597  3020 

30.62 

1.608  2820 

30.38 

1.619  !77Z 

30.15 

1.629  9945 

29.95 

6 

7 

•597  4857 
•597  6694 

30.62 
30.62 

.608  4642 
.608  6465 

30-38 
30-37 

.619  3581 
.619  5390 

30-I5 
30.15 

.630  1742 
.630  3538 

29.94 
29.94 

8 

•597  8531 

30.61 

.608  8287 

30-37 

.619  7199 

30.14 

.630  5335 

29.94 

9 

.598  0368 

30.61 

.609  0109 

30.36 

.619  9007 

30-I4 

.630  7131 

29.93 

10 

1.598  2204 

30.60 

1.609  1931 

30.36 

1.620  0816 

30.14 

1.630  8927 

29.93 

11 

.598  4040 

30.60 

.609  3752 

3°-36 

.620  2623 

30.13 

.631  0722 

29.93 

12 

.598  5876 

30.59 

.609  5573 

30-35 

.620  4431 

30.I3 

.631  2518 

29.92 

13 

.598  7711 

3°-59 

.609  7394 

3°-35 

.620  6239 

30.12 

•631  43J3 

29.92 

14 

.598  9547 

3°-59 

.609  9215 

30-34 

.620  8046 

30.12 

.631  6108 

29.92 

15 

1.599  '382 

30.58 

1.610  1036 

30-34 

1.620  9853 

30.12 

1.631  7903 

29.91 

16 

•599  3217 

30.58 

.610  2856 

30.34 

.621  1660 

30.11 

.631  9698 

29.91 

17 

•599  5051 

3°-57 

.610  4676 

30-33 

.621  3467 

30.11 

.632  1492 

29.91 

18 

.599  6885 

3°-57 

.610  6496 

30.33 

.621  5274 

30.11 

.632  3286 

29.90 

19 

•599  87i9 

3°-57 

.610  8315 

30-32 

.621  7080 

30.10 

.632  5081 

29.90 

20 

i.  600  0553 

30.56 

1.611  0135 

30.32 

1.621  8886 

30.10 

1.632  6875 

29.90 

21 

.600  2387 

30.56 

.611  1954 

30.32 

.622  0692 

3P.IO 

.632  8668 

29.89 

22 

.600  4220 

3°-55 

.611  3773 

30-31 

.622  2497 

30.09 

.633  0462 

29.89 

23 

.600  6053 

30.55 

.611  5591 

3°-3' 

.622  4307 

3^-09 

.633  225C 

29.89 

24 

.600  7886 

30-55 

.611  7410 

30.31 

.622  6108 

30.09 

.633  4048 

29.88 

25 

1.600  9718 

3°-54 

1.611  9228 

30-3° 

1.622  7917 

30.08 

1.633  5841 

29.88 

26 

.601  1551 

3°-54 

.612  1046 

30.30 

.622  9718 

30.08 

.633  7634 

29.88 

27 

.601  3383 

3°-53 

.612  2864 

30.29 

.623  1523 

30.08 

.633  9427 

29.87 

28 

.601  5214 

3°-53 

.612  4681 

30.29 

.623  3327 

3.0.07 

.634  1219 

29.87 

29 

.601  7046 

30.52 

.612  6499 

30.29 

-623  5131 

30.07 

.634  3011 

29.87 

30 

1.601  8877 

30.52 

i.  612  8316 

30.28 

*-623  6935 

30.06 

1.634  4803 

29.86 

31 

.602  0708 

30-52 

.613  0132 

30.28 

.623  8739 

30.06 

-634  6595 

29.86 

32 

.602  2539 

30-51 

.613  1949 

30.28 

.624  0543 

30.06 

.634  8387 

29.86 

33 

.602  4370 

30.51 

.613  3765 

30.27 

.624  2346 

30.05 

.635  0178 

29.86 

34 

.602  6200 

30.50 

.613  5582 

30.27 

.624  4149 

30,05 

.635  1969 

29.85 

35 

1.602  8030 

30.50 

1.613  7398 

30.26 

i.6z4  5952 

30.05 

1.635  376o 

29.85 

36 

.602  9860 

30.50 

.613  9213 

30.26  , 

.624  7755 

30.04 

•635  555i 

29.85 

37 

38 
39 

.603  1690 
.603  3519 
.603  5348 

30.49 

30.49 

30.48 

.614  1029 
.614  2844 
.614  4659 

30.26 
30.25 
30.25 

.624  9557 
.625  1360 
.625  3162 

30.04 
30.04 
30.03 

.635  7342 
.635  9132 
.636  0922 

29.84 
29.84 
29.84 

40 

1.603  7»77 

30.48 

1.614  6474 

30.25 

1.625  4964 

30.03 

1.636  2713 

29.83 

41 
42 
43 
44 

.603  9005 
.604  0834 
.604  2662 
.604  4490 

3°-47 
30.47 
30.47 
30.46 

.614  8288 
.615  0103 
.615  1917 
•615  3731 

30.24 
30.24 

30-23 
30-23 

.625  6765 
.625  8567 
.626  0368 
.626  2169 

30.03 
30.02 
30.02 
30.02 

.636  4502 
.636  6292 
.636  8082 
.636  9871 

29.83 
29.83 
29.82 
29.82 

45 

1.604  6317 

30.46 

1-615  5545 

30.23 

1.626  3970 

30.01 

1.637  1660 

29.82 

46 

.604  8145 

3°-45 

.615  7358 

30.22 

.626  5771 

30.01 

•637  3449 

29.82 

47 
48 

.604  9972 
.605  1799 

3°-45 
3°-45 

.615  9171 
.6iv6  0984 

30.22 
30.22 

.626  7571 
.626  9372 

30.01 
30.00 

.637  5238 
.637  7027 

29.81 
29.81 

49 

.605  3626 

3°-44  , 

.616  2797 

30.21 

.627  1172 

30.00 

.637  8815 

29.81 

50 

1.605  5452 

3°-44 

1.616  4610 

30.21 

1.627  2972 

30.00 

1.638  0603 

29.80 

51 

.605  7278 

3°-43 

.616  6422 

30.20 

.627  4771 

29.99 

.638  2391 

29.80 

52 

.605  9104 

3°-43 

.616  8234 

30.20 

.627  6571 

29.99 

.638  4179 

29.80 

53 

.606  0930 

30.43  , 

.617  0046 

30.20 

.627  8370 

29.99 

.638  5967 

29-79 

54 

.606  2755 

30.42 

.617  1858 

30.19 

.628  0169 

29.98 

.638  7754 

29.79 

55 
56 
57 

58 
59 

i.  606  4581 
.606  6406 
.606  8230 
.607  0055 
.607  1879 

30.42 

30-42 
30.41 
30.41 
30.40 

1.617  3669 
.617  5481 
.617  7292 
.617  9102 
.618  0913 

30.19 
30.19 
30.18 
30.18 
30.17 

1.628  1968 
.628  3766 
.628  5565 
.628  7363 
.628  9161 

29.98 
2998 
29.97 
29.97 
29.97 

1.638  9542 
.639  1329 
.639  3116 
.639  4902 
.639  6689 

29.79 
29.78 
29.78 
29.78 
29.77 

60 

1.607  3703 

30.40 

1.618  2724 

30.17 

1.629  °959 

29.96 

1.639  8475 

29.77 

579 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

56° 

57° 

58° 

59° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0' 

1.639  8475 

29.77 

1.650  5336 

49.60 

1.661  1601 

49.44 

1.671  7331 

493o 

1 

.640  0262 

29.77 

.650  7112 

49.60 

.661  3368 

49.44 

.671  90^9 

49.30 

2 

.640  2048 

29.77 

.650  8887 

29.59 

.661  5134 

49.44 

.674  0846 

49.30 

3 

.640  3833 

29.76 

.651  0663 

29.59 

.661  6900 

29.43 

.674  4604 

29.29 

4 

.640  5619 

29.76 

.651  4438 

49.59 

.661  8666 

49.43 

.672  4362 

29.29 

5 

1.640  7405 

29.76 

1.651  4213 

49.58 

1.662  0432 

29.43 

1.672  6119 

29.29 

6 

.640  9190 

29.75 

.651  5988 

49.58 

.662  2197 

29-43 

.672  7876 

29.29   1 

7 

.641  0975 

29.75 

.651  7763 

29.58 

.662  3963 

29.42 

.672  9634 

29.28 

8 

.641  2760 

29.75 

•651  9538 

49.58 

.662  5728 

29.42 

.673  1391 

29.28 

9 

.641  4545 

29.74 

.652  1312 

29.57 

.662  7493 

29.42 

-673  3*47 

29.28 

10 

1.641  6329 

29.74 

1.652  3086 

29-57 

1.662  9258 

49.42 

1.673  4904 

29.28 

11 

.641  8114 

29.74 

.652  4861 

49.57 

.663  1023 

29.41 

.673  6661 

29.28 

12 

.641  9898 

29.74 

.652  6635 

49.57 

.663  2788 

29.41 

.673  8417 

29.27 

13 

.642  1682 

29.73 

.652  8408 

29.56 

•663  4553 

29.41 

.674  0174 

29.27 

14 

.642  3466 

29-73 

.653  0182 

29.56 

.663  6317 

49.41 

.674  1930 

29.47 

15 

1.642  5250 

29.73 

1.653  1956 

49.56 

1.663  8o82 

49.40 

1.674  3686 

49.47 

16 

.642  7033 

29.72 

.653  3729 

29-55 

.663  9846 

49.40 

.674  5442 

49.47 

17 

.642  8816 

29.72 

.653  5502 

29-55 

.664  1610 

49.40 

.674  7198 

29.26 

18 

.643  0599 

29.72 

.653  7275 

29-55 

.664  3374 

49.40 

.674  8954 

29.26 

19 

.643  2382 

29.71 

.653  9048 

29-55 

.664  5137 

29-39 

.675  0709 

29.26 

20 

1.643  4165 

29.71 

1.654  0821 

49.54 

1.664  6901 

29-39 

1.675  2465 

29.26 

21 

.643  5948 

29.71 

.654  2593 

49.54 

.664  8664 

29-39 

.675  4220 

2925 

22 

.643  7730 

29.71 

.654  4366 

29.54 

.665  0428 

29-39 

•675  5975 

29.25 

23 

.643  9513 

29.70 

.654  6138 

29.54 

.665  2191 

29-39 

.675  7730 

29.25 

24 

.644  1295 

29.70 

.654  7910 

29-53 

.665  3954 

49.38 

.675  9485 

29.25 

25 

1.644  3°77 

29.70 

1.654  9682 

29-53 

1.665  5717 

49.38 

1.676  1240 

29.25 

26 

.644  4858 

49:69 

•655  H54 

29.53 

.665  7480 

49.38 

.676  2995 

29.24 

27 

.644  6640 

29/69 

•655  3225 

29-53 

.665  9242 

49.38 

.676  4749 

29.24 

28 

.644  8421 

29.69 

•655  4997 

49.54 

.666  1005 

49.37 

.676  6504 

29.24 

29 

.645  0203 

29.69 

.655  6768 

29.52 

.666  2767 

29-37 

.676  8258 

29.24 

30 

1.645  J9^4 

29.68 

*-655  ^539 

49.54 

1.666  4529 

29-37 

1.677  0012 

29.24 

31 

•645  3765 

29.68 

.656  0310 

49.51 

.666  6291 

29-37 

.677  1766 

29.23 

32 

•645  5545 

29.68 

.656  2081 

4951 

.666  8053 

49.36 

.677  3520 

29-23 

33 

.645  7326 

29.67 

.656  3852 

49.51 

.666  9815 

29.36 

.677  5274 

29.23 

34 

.645  9106 

29,67 

.656  5644 

49.51 

.667  1577 

2936 

.677  7028 

29.23 

35 

1.646  0886 

49.67 

1.656  7394 

49.50 

1.667  3338 

29.36 

1.677  878I 

29.23 

36 

.646  2  6  #6 

29.67 

.656  9163 

49.50 

.667  5100 

29-35 

.678  0535 

29.22 

37 

.646  4446 

2,9.66 

•*57  °933 

29.50 

.667  6861 

29-35 

.678  2488 

29.22 

38 

.646  6226 

29.66 

.657  4703 

29.50 

.667  8622 

29-35 

.678  4041 

29.22 

39 

.646  8005 

49.66 

.657  4474 

29.49 

.668  0383 

29-35 

•678  5794 

29.22 

40 

1.646  9785 

29.65 

1.657  6242 

49.49 

1.668  2144 

29-35 

1.678  7547 

29.22 

41 

.647  1564 

29.65 

.657  8011 

49.49 

.668  3904 

29.34 

.678  9300 

29.21 

42 

•647  3343 

29.65 

.657  9781 

29.49 

.668  5665 

29.34 

.679  1051 

29.21 

43 

.647  5122 

29.65 

.658  '1550 

29.48 

.668  7425 

29-34 

.679  2806 

29.21 

44 

.647  6900 

29.64 

.658  3318 

49.48 

.668  9185 

29.34 

.679  4558 

29.21 

45 

1.647  8679 

29.64 

1.658  5087 

49.48 

1.669  °945 

29-33 

1.679  6310 

29.20 

46 

.648  0457 

29.64 

.658  6855 

49.48 

.669  2705 

29-33 

.679  8063 

29.20 

47 

.648  2235 

29.63 

.658  8624 

29.47 

.669  4465 

29-33 

.679  9815 

29.20 

48 

.648  4013 

29.63 

.659  0393 

49.47 

.669  6225 

29-33 

.680  1567 

29.20 

49 

.648  5791 

29.63 

.659  2161 

49.47 

.669  7984 

29.32 

.680  3319 

29.20 

50 

1.648  7569 

29.63 

1.659  3929 

49.47 

1.669  9744 

29.32 

i.  680  5070 

29.19 

51 

.648  9346 

29.62 

.659  5697 

49.46 

.670  1503 

29.32 

.680  6822 

29.19 

52 

.649  1123 

29.62 

.659  7465 

49.46 

.670  3262 

29.32 

.680  8574 

29.19 

53 

.649  2901 

29.62 

.659  9232 

49.46 

.670  5021 

29.32 

.681  0325 

29.19 

54 

.649  4677 

29.61 

.660  1000 

49.46 

.670  6780 

29.31 

.681  2076 

29.19 

55 

1.649  6454 

29.61 

1.  660  2767 

49.45 

1.670  8539 

29.31 

1.681  3827 

29.18 

56 

.649  8231 

29.61 

.660  4534 

49.45 

.671  0298 

49.31 

.681  5578 

29.18 

57 

.650  0007 

29.61 

.660  6301 

49.45 

.671  2056 

49.31 

.681  7329 

29.18 

58 

.650  1784 

29.60 

.660  8068 

49.45 

.671  3814 

49.30 

.681  9080 

29.18 

59 

.650  3560 

29.60 

.660  9835 

29.44 

•671  5573 

49.30 

.682  0831 

29.18 

60 

1.650  5336 

29.60 

1.661  1601 

49.44 

1.671  7331 

49.30 

1.684  4581 

29.17 

580 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

60° 

61° 

62° 

63° 

log  M. 

Diff.  1". 

logM 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

O' 

1.682  2581 

29.17 

1.692  7408 

29.07 

1.703  1866 

28.97 

1.713  6006 

28.89 

1 

2 

.682.  4332, 
.682  6082 

29.17 
29.17 

.692  9152 
.693  0896 

29.06 
29.06 

.703  3604 
.703  5342 

28.97 
28.97 

•713  7739 
•713  9473 

28.89 
28.89 

3 

.682  7832 

29.17 

.693  2640 

29.06 

.703  7080 

28.97 

.714  1206 

28.88 

4 

.682  9582 

29.17 

•693  4383 

29.06 

.703  8818 

28.96 

.714  2939 

28.88 

5 

1.683  1332 

29.16 

1.693  6127 

29.06 

1.704  0556 

28.96 

1.714  4672 

28.88 

6 

.683  3082 

29.16 

.693  7870 

29.05 

.704  2293 

28.96 

.714  6405 

28.88 

7 

.683  4832 

29.16 

.693  9613 

29.05 

.704  4031 

28.96 

.714  8138 

28.88 

8 

.683  6581 

29.16 

.694  1356 

29.05 

.704  5768 

28.96 

.714  9870 

28.88 

9 

.683  8331 

29.16 

.694  3099 

29.05 

.704  7506 

28.96 

.715  1603 

28.88 

10 

1.684  0080 

29.16 

1.694  4842 

29.05 

1.704  9243 

28.96 

1.715  3336 

28.88 

11 

.684  1830 

29.15 

.694  6585 

29.04 

.705  0981 

28.95 

.715  5068 

28.88 

12 

.684  3579 

29.15 

.694  8328 

29.04 

.705  2718 

28.95 

.715  6801 

28.87 

13 

.684  5328 

29.15 

.695  0070 

2904 

•7°5  4455 

28.95 

•7*5  8533 

28.87 

14 

.684  7077 

29.15 

.695  1813 

29.04 

.705  6192 

28.95 

.716  0266 

28.87 

15 

1.684  8826 

29.14 

'•695  3555 

29.04 

1.705  7929 

28.95 

1.716  1998 

28.87 

16 

.685  0574 

29.14 

.695  5298 

29.04 

.705  9666 

28.95 

.716  3730 

28.87 

17 

.685  2323 

29.14 

.695  7040 

29.04 

.706  1402 

28.95 

.716  5462 

28.87 

18 

.685  4071 

29.14 

.695  8782 

29.03 

.706  3139 

28.94 

.716  7194 

28.87 

19 

.685  5820 

29.14 

.696  0524 

29.03 

.706  4875 

28.94 

.716  8926 

28.87 

20 

1.685  7568 

29.14 

1.696  2266 

29.03 

1.706  6612 

28.94 

1.717  0658 

28.86 

21 

.685  9316 

29.13 

.696  4008 

29.03 

.706  8348 

28.94 

.717  2390 

28.86 

22 

.686  1064 

29.13 

.696  5750 

29.03 

.707  0085 

28.94 

.717  4122 

28.86 

23 

.686  2812 

29.13 

.696  7491 

29.03 

.707  1821 

28.94 

•7X7  5853 

28.86 

24 

.686  4560 

29.13 

.696  9233 

29.02 

•707  3557 

28.94 

.717  7585 

28.86 

25 

1.686  6308 

29.13 

1.697  0974 

29.02 

1.707  5293 

28.93 

1.717  9317 

28.86 

26 

.686  8055 

29.13 

.697  2716 

29.02 

.707  7029 

28.93 

.718  1048 

28.86 

27 

.686  9803 

29.12 

•697  4457 

29.02 

.707  8765 

28.93 

.718  2780 

28.86 

28 

.687  1550 

29.12 

.697  6198 

29.02 

.708  0501 

28.93 

.718  4511 

28.86 

29 

.687  3297 

29.12 

.697  7939 

29.02 

.708  2237 

28.93 

.718  6242 

28.85 

30 

1.687  5044 

29.12 

1.697  9680 

29.02 

1.708  3972 

28.93 

1.718  7974 

28.85 

31 

.687  6791 

29.12 

.698  1421 

29.01 

.708  5708 

2893 

.718  9705 

28.85 

32 

.687  8538 

29.11 

.698  3162 

29.01 

.708  7444 

28,92 

.719  1436 

28.85 

33 

.688  0285 

29.11 

.698  4902 

29.01 

.708  9179 

28.92 

.719  3167 

28.85 

34 

.688  2032 

29.11 

.698  6643 

29.01 

.709  0914 

28.92 

.719  4898 

28.85 

35 

1.688  3778 

29.11 

1.698  8383 

29.01 

1.709  2650 

28.92 

1.719  6629 

28.85 

36 

.688  5525 

29.11 

.699  0124 

29.01 

.709  4385 

28.92 

.719  8360 

28.85 

37 

.688  7271 

29.10 

.699  1864 

29.00 

.709  6120 

28.92 

.720  0090 

28.85 

38 

.688  9017 

29.10 

.699  3604 

29.00 

.709  7855 

28.92 

.720  1821 

28.84 

39 

.689  0764 

29.10 

.699  5345 

29.00 

.709  9590 

28.92 

.720  3552 

28.84 

40 

1.689  2510 

29.10 

1.699  7085 

29.00 

1.710  1325 

28.91 

1.720  5282 

28.84 

41 

.689  4256 

29.10 

.699  8824 

29.00 

.710  3060 

28.91 

.720  7013 

28.84 

42 

.689  6001 

29.09 

.700  0564 

29.00 

.710  4794 

28.91 

.720  8743 

28.84 

43 

.689  7747 

29.09 

.700  2304 

29.00 

.710  6529 

28.91 

.721  0474 

28.84 

44 

.689  9493 

29.09 

.700  4044 

28.99 

.710  8263 

28.91 

.721  2204 

28.84 

45 

1.690  1238 

29.09 

1.700  5783 

28.99 

1.710  9998 

28.91 

1.721  3934 

28.84 

46 

47 

.690  2984 
.690  4729 

29.09 
29.09 

.700  7523 
.700  9262 

28.99 
28.99 

.711  1732 
.711  3467 

28.91 
28.90 

.721  5665 
.721  7395 

28.84 
28.84 

48 

.690  6474 

29.09 

.701  looi 

28.99 

.711  5201 

28.90 

.721  9125 

28.83 

49 

.690  8219 

29.08 

.701  2741 

28.99 

.711  6935 

28.90 

.722  0855 

28.83 

50 

1.690  9964 

29.08 

1.701  4480 

28.98 

1.711  8669 

28.90 

1.722  2585 

28.83 

51 

.691  1709 

29.08 

.701  6219 

28.98 

.712  0403 

28.90 

.722  4315 

28.83 

52 

.691  3454 

29.08 

.701  7958 

28.98 

.712  2137 

28.90 

.722  6044 

28.83 

53 

.691  5199 

29.08 

.701  9697 

28.98 

.712  3871 

28.90 

.722  7774 

28.83 

54 

.691  6943 

29.08 

.702  1435 

28.98 

.712  5605 

28.90 

.722  9504 

28.83 

55 

1.691  8688 

29.07 

1.702  3174 

28.98 

1.712  7339 

28.90 

1.723  1233 

2^3 

56 

.692  0432 

29.07 

.702  4913 

28.98 

.712  9072 

28.89 

.723  2963 

28.83 

57 

.692  2176 

29.07 

.702  6651 

28.97 

.713  0806 

28.89 

.723  4693 

28.82 

58 

.692  3920 

29.07 

.702  8389 

28.97 

•713  2539 

28.89 

.723  6422 

28.82 

59 

.692  5664 

29.07 

.703  0128 

28.97 

•713  4*73 

28.89 

.723  8151 

28.82 

60 

1.692  7408 

29.07 

1.703  1866 

22.97 

1.713  6006 

28.89 

1.723  9881 

28.82 

581 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

64° 

65° 

66° 

67° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0' 

1.723   9881 

28.82 

1-734  3539 

28.77 

1.744  7°31 

28.73 

1-755  °4°5 

28.70 

1 

.724  1610 

28.82 

•734  5265 

28.77 

•744  8755 

28.73 

•755  2127 

28.70 

2 

•724  3339 

28.82 

•734  699i 

28.77 

•745  °479 

28.73 

•755  3849 

28.70 

3 

.724  5068 

28.82 

•734  8718 

28.77 

.745  2202 

28.73 

•755  557i 

28.70 

4 

.724  6798 

28.82 

•735  °444 

28.77 

•745  3926 

28.73 

•755  7293 

28.70 

5 

1.724  8527 

28.82 

1.735  2169 

28.76 

1.745  565o 

28.73 

1.755  90i5 

28.70 

6 

.725  0256 

28.82 

•735  3895 

28.76 

•745  7373 

28.73 

.756  0737 

28.70 

7 

.725  1984 

28.81 

.735  5621 

28.76 

•745  9°97 

28.73 

.756  2459 

28.70 

8 

.725  3713 

28.81 

•735  7347 

28.76 

.746  0820 

28.72 

.756  4181 

28.70 

9 

.725  5442 

28.81 

•735  9°73 

28.76 

.746  2544 

28.72 

•756  59°3 

28.70 

10 

1.725  7171 

28.81 

1.736  0798 

28.76 

1.746  4267 

28.72 

1.756  7625 

28.70 

11 

.725  8900 

28.81 

.736  2524 

28.76 

.746  5991 

28.72 

•756  9347 

28.70 

12 

.726  0628 

28.81 

.736  4250 

28.76 

.746  7714 

28.72 

•757  1069 

28.70 

13 

.726  2357 

28.81 

•736  5975 

28.76 

.746  9437 

28.72 

•757  2791 

28.70 

14 

.726  4085 

28.81 

.736  7701 

28.76 

.747  1161 

28.72 

•757  4513 

28.70 

15 

1.726  5814 

28.81 

1.736  9426 

28.76 

1.747  2884 

28.72 

1.757  6235 

28.70 

16 

.726  7542 

28.81 

•737   "52 

28.76 

.747  4607 

28.72 

•757  7957 

28.70 

17 

.726  9270 

28.81 

•737  2877 

28.76 

•747  6330 

28.72 

•757  9679 

28.70 

18 

.727  0999 

28.80 

.737  4602 

28.76 

.747  8054 

28.72 

.758   1401 

28.70 

19 

.727  2727 

28.80 

.737  6328 

28.75 

•747  9777 

28.72 

.758  3123 

28.70 

20 

1-727  4455 

28.80 

1.737  8053 

28.75 

1.748  1500 

28.72 

1.758  4844 

28.70 

21 

.727  6183 

28.80 

•737  9778 

28.75 

.748  3223 

28.72 

.758  6566 

28.70 

22 

.727  7911 

28.80 

•738   i5°3 

28.75 

.748  4946 

28.72 

.758  8288 

28.70 

23 

.727  9639 

28.80 

.738   3228 

28.75 

.748  6669 

28.72 

.759  ooio 

28.70 

24 

.728  1367 

28.80 

•738  4953 

28.75 

.748  8392 

28.72 

•759  1731 

28.70 

25 

1.728  3095 

28.80 

1.738  6679 

28.75 

1.749  °"5 

28.72 

'•759  3453 

28.70 

26 
27 

.728  4823 
.728  6551 

28.80 
28.80 

.738  8404 
.739  0129 

28.75 
28.75 

•749  l838 
•749  3561 

28.72 
28.72 

•759  5175 
•759  6897 

28.70 
28.70 

28 

.728  8279 

28.80 

•739  1857 

28.75 

•749  5284 

28.72 

.759  8618 

28.69 

29 

.729  0006 

28.79 

•739  3578 

28.75 

•749  70°7 

28.71 

.760  0340 

28.69 

30 

1.729  1734 

28.79 

1-739  53°3 

28.75 

1-749  873° 

28.71 

1.760  2062 

28.69 

31 

.729  3461 

28.79 

•739  7°28 

28.75 

•75°  °453 

28.71 

.760  3783 

28.69 

32 

.729  5189 

28.79 

•739  8753 

28.75 

.750  2176 

28.71 

.760  5505 

28.69 

33 

.729  6916 

28.79 

.740  0477 

28.75 

•75°  3898 

28.71 

.760  7227 

28.69 

34 

.729  8644 

28.79 

.740    2202 

28.74 

.750  5621 

28.71 

.760  8948 

28.69 

35 

1.730  0371 

28.79 

1-74°    3927 

28.74 

1.750  7344 

28.71 

1.761  0670 

28.69 

36 

.730  2099 

28.79 

•74°  565i 

28.74 

.750  9067 

28.71 

.761  2392 

28.69 

37 

.730  3826 

28.79 

.740  7376 

28.74 

.751  0789 

28.71 

.761  4113 

28.69 

38 

•73°  5553 

28.79 

.740  9101 

28.74 

.751  2512 

28.71 

.761  5835 

28.69 

39 

.730  7280 

28.79 

.741  0825 

28.74 

•751  4234 

28.71 

.761  7556 

28.69 

40 

1.730  9007 

28.78 

1.741   2550 

28.74 

1-75'  5957 

28.71 

1.761  9278 

28.69 

41 

.731  0735 

28.78 

.741  4274 

28.74 

.751  7680 

28.71 

.762  0999 

28.69 

42 

.731  2462 

28.78 

.741   5998 

28-74 

.751  9402 

28.71 

.762  2721 

28.69 

43 

.731  4189 

28.78 

.741  7723 

28.74 

.752  1125 

28.71 

.762  4442 

28.69 

44 

•731   5915 

28.78 

•741   9447 

28.74 

.752  2847 

28.71 

.762  6164 

28.69 

45 

1.731   7642 

28.78 

1.742  1171 

28.74 

1.752  4570 

28.71 

1.762  7885 

28.69 

I    46 

.731   9369 

28.78 

.742  2896 

28.74 

.752  6292 

28.71 

.762  9607 

28.69 

47 

.732  1096 

28.78 

.742  4620 

28.74 

.752  8015 

18.71 

.763  1328 

28.69 

48 

.732  2823 

28.78 

.742  6344 

28.74 

•752  9737 

a8.7i 

.763  3050 

28.69 

49 

•732  4549 

28.78 

.742  8068 

28.74 

•753   H6° 

28.71 

.763  4771 

28.69 

50 

1.732  6276 

28.78 

1.742  9792 

28.74 

1-753  3l82 

28.71 

1.763  6493 

28.69 

51 

.732  8002 

28.78 

•743  *5i6 

28.73 

•753  49°4 

28.71 

.763  8214 

28.69 

1     52 

.732  9729 

28.77 

•743   324° 

28.73 

•753  6627 

28.71 

.763  9936 

28.69 

53 

•733   "455 

28.77 

•743  4964 

28.73 

•753  8349 

28.71 

.764  1657 

28.69 

54 

•733  3'82 

28.77 

.743  6688 

28.73 

•754  °°7i 

28.70 

•764  3379 

28.69 

55 

1-733  49°8 

28.77 

1.743  8412 

28.73 

1-754  1794 

28.70 

1.764  5100 

28.69 

56 

•733  6635 

28.77 

.744  0136 

28.73 

•754  35l6 

28.70 

.764  6821 

28.69 

57 

58 

•733  836i 
•734  °°87 

28.77 
28.77 

.744  1860 
.7^4  3584 

28.73 
28.73 

•754  5238 
.754  6960 

28.70 
28.70 

.764  8543 
.765  0264 

28.69 
28.69 

59 

•734  l8l3 

28.77 

•744  53°8 

28.73 

.754  8682 

28.70 

.765  1985 

28.69 

60 

1-734  3539 

28.77 

1.744  703! 

28.73 

1.755  0405 

28.70 

1.765   3707 

28.69 

582 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

68° 

69° 

70° 

71° 

logM. 

Diflf.  1". 

logM. 

Diflf.  1". 

logM. 

Diflf.  1". 

logM. 

Diflf.  1". 

0 

1.765  3707 

28.69 

•775  6985 

28.69 

1.786  0284 

28.70 

1.796  3650 

28.73 

1 

.765  5428 

28.69 

•775  8706 

28.69 

.786  2006 

28.70 

.796  5374 

28.73 

2 

.765  7150 

28.69 

.776  0427 

28.69 

.786  3728 

28.70 

.796  7097 

28.73 

3 

.765  8871 

28.69 

.776  2149 

28.69 

.786  5450 

28.70 

.796  8821 

28.73 

4 

.766  0592 

28.69 

.776  3870 

28.69 

.786  7172 

28.70 

•797  0545 

28.73 

5 

1.766  2314 

28.69 

.776  5591 

28.69 

1.786  8894 

28.70 

1.797  2268 

28.73 

6 

.766  4035 

28.69 

.776  7313 

28.69 

.787  0617 

28.70 

•797  3992 

28.73 

7 

.766  5756 

28.69 

.776  9034 

28.69 

•787  2339 

28.70 

•797  57»6 

28.73 

8 

.766  7478 

28.69 

•777  0755 

28.69 

.787  4061 

28.70 

•797  744° 

28.73 

9 

.766  9199 

28.69 

•777  *477 

28.69 

•787  5783 

28.70 

•797  9l64 

28.73 

10 

1.767  0920 

28.69 

1.777  4198 

28.69 

1.787  7506 

28.70 

1.798  0888 

28.73   i 

11 

.767  2642 

28.69 

•777  5920 

28.69 

.787  9228 

28.71 

.798  2611 

28.73 

12 

.767  4363 

28.69 

.777  7641 

28.69 

.788  0950 

28.71 

•798  4335 

28.73 

13 

.767  6084 

28.69 

•777  9363 

28.69 

•788  2673 

28.71 

.798  6060 

28.73 

14 

.767  7805 

28.69 

.778  1084 

28.69 

.788  4395 

28.71 

•798  7784 

28.73 

15 

1.767  9527 

28.69 

1.778  2806 

28.69 

1.788  6117 

28.71 

1.798  9508 

28.73 

16 

.768  1248 

28.69 

.778  4527 

28.69 

.788  7840 

28.71 

.799  1232 

28.74 

17 

.768  2969 

28.69 

.778  6248 

28.69 

.788  9562 

28.71 

•799  2956 

28.74 

18 

.768  4691 

28.69 

.778  7970 

28.69 

.789  1284 

28.71 

.799  4680 

28.74 

19 

.768  6412 

28.69 

•778  969' 

28.69 

.789  3007 

28.71 

.799  6404 

28.74 

20 

I.768  8133 

28.69 

1.779  1413 

28.69 

1.789  4730 

28.71 

1.799  8128 

28.74 

21 

.768  9854 

28.69 

•779  3*4° 

28.69 

.789  6452 

28.71 

•799  9853 

28.74 

22 

.769  1576 

28.69 

.779  4862 

28.69 

•789  8175 

28.71 

.800  1577 

28.74 

23 

.769  3297 

28.69 

•779  6578 

28.69 

•789  9897 

28.71 

.800  3301 

28.74 

24 

.769  5018 

28.69 

•779  8299 

28.69 

.790  1620 

28.71 

.800  5026 

28-74 

25 
26 

1.769  6740 
.769  8461 

28.69 
28.69 

1.780  OO2I 
.780  1742 

28.69 
28.69 

1.790  3342 
.790  5065 

28.71 
28.71 

1.800  6750 
.800  8475 

28.74 
28.74 

27 

.770  0182 

28.69 

.780  3464 

28.69 

.790  6788 

28.71 

.801  0199 

28.74 

28 

.770  1903 

28.69 

.780  5185 

28.69 

•79°  8510 

28.71 

.801  1924 

28.74 

29 

.770  3625 

28.69 

.780  6907 

28.69 

.791  0233 

28.71 

.801  3648 

28.74 

30 

1.770  5346 

28.69 

1.780  8629 

28.69 

1.791  1956 

28.71 

1.801  5373 

28.74 

31 

.770  7067 

28.69 

.781  0350 

28.69 

.791  3678 

28.71 

.801  7107 

28.74 

32 

.770  8788 

28.69 

.781  2072 

28.69 

.791  5401 

28.71 

.801  8822 

28.74 

33 

.771  0510 

28.69 

.781  3793 

28.69 

.791  7124 

28.71 

.802  0547 

28.75 

34 

.771  2231 

28.69 

.781  5515 

28.69 

.791  8847 

28.71 

.802  2271 

28.75 

35 

1.771  3952 

28.69 

1.781  7237 

28.69 

1.792  0570 

28.71 

1.802  3996 

28.75 

36 
37 

.771  5673 
•77*  7395 

28.69 
28.69 

.781  8959 
.782  0680 

28.69 
28.70 

.792  2293 
.792  4016 

28.71 
28.72 

.802  5721 
.802  7446 

28.75 
28.75 

38 

.771  9116 

28.69 

.782  2402 

28.70 

.792  5738 

28.72 

.802  9171 

28.75 

39 

.772  0837 

28.69 

.782  4124 

28.70 

.792  7461 

28.72 

.803  0896 

28.75 

40 

1.772  2559 

28.69 

1.782  5845 

28.70 

1.792  9184 

28.72 

1.803  2621 

28.75 

41 

.772  4280 

28.69 

.782  7567 

28.70 

•793  °9°7 

28.72 

.803  4346 

28.75 

42 

.772  6001 

28  69 

.782  9289 

28.70 

•793  263° 

2.8.72 

.803  6071 

28.75 

43 

.772  7722 

28.69 

.783  ion 

28.70 

•793  4354 

28.72 

.803  7796 

28.75 

44 

.772  9444 

28.69 

.783  2732 

28.70 

•793  6°77 

28.72 

.803  9521 

28.75 

45 

1.773  1165 

28.69 

1-783  4454 

28.70 

1.793  7800 

28.72 

1.804  1246 

28.75 

46 

.773  2886 

28.69 

.783  6176 

28.70 

•793  9523 

28.72 

.804  2971 

28.75 

47 
1  48 

•773  4607 
•773  6329 

28.69 
28.69 

.783  7898 
.783  9620 

28.70 
28.70 

.794  1246 
•794  2969 

28.72 
28.72 

.804  4697 
.804  6422 

28.75 
28.76 

49 

•773  8°5° 

28.69 

.784  1342 

28.70 

.794  4693 

28.72 

.804  8147 

28.76 

50 
51 

1773  9771 
.774  1493 

28.69 
28.69 

1.784  3064 
.784  4786 

28.70 
28.70 

1.794  6416 
•794  8139 

28.72 
28.72 

1.804  9877 
.805  1598 

28.76 
28.76 

52 
53 

.774  3214 
•774  4935 

28.69 
28.69 

.784  6508 
.784  8230 

28.70 
28.70 

•794  9862 
•795  1586 

28.72 
28.72 

-805  3324 
•805  5049 

28.76 
28.76 

54 

•774  6657 

28.69 

.784  9952 

28.70 

•795  33°9 

28.72 

•805  6775 

28.76 

55 
56 

1-774  8378 
.775  0099 

28.69 
28.69 

1.785  1674 
•785  3396 

28.70 
28.70 

1-795  5°33 
•795  6756 

28.72 
28.72 

1.805  8500 
.806  0226 

28.76 
28.76 

57 

.775  1821 

28.69 

.785  5118 

28.70 

.795  8480 

28.72 

.806  1952 

28.76 

58 

•775  3542 

28.69 

.785  6840 

28.70 

.796  0203 

28.73 

.806  3677 

28.76   1 

59 

•775  5263 

28.69 

.785  8562 

28.70 

.796  1927 

28.73 

.806  5403 

28.76 

60 

1.775  6985 

28.69 

1.786  0284 

28.70 

1.796  3650 

28.73 

i.  806  7129 

28.76 

583 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

72° 

73° 

74° 

75° 

log  It, 

Difif.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0' 

i.  806  7129 

28.76 

1.817  0765 

28.81 

1.827  4602 

28.88 

1.837  8686 

28.95 

1 

.806  8855 

28.76 

.817  2494 

28.81 

.827  6335 

28.88 

.838  0423 

28.95 

2 

.807  0581 

28.77 

.817  4222 

28.82 

.827  8068 

28.88 

.838  2160 

28.95 

3 

.807  2307 

28.77 

.817  5951 

28.82 

.827  9800 

28.88 

.838  3898 

28.95 

4 

.807  4033 

28.77 

.817  7680 

28.82 

.828  1533 

28.88 

•838  5635 

28.96 

5 

'•8°7  5759 

28.77 

1.817  9410 

28.82 

1.828  3266 

28.88 

1.838  7372 

28.96 

6 

.807  7485 

28.77 

.818  1139 

28.82 

.828  4999 

28.88 

.838  9110 

28.96 

7 

.807  9211 

28.77 

.818  2868 

28.82 

.828  6732 

28.88 

.839  0847 

28.96 

8 

.808  0937 

28.77 

.818  4597 

28.82 

.828  8465 

28.88 

.839  2585 

28.96 

9 

.808  2663 

28.77 

.818  6326 

28.82 

.829  0198 

28.89 

.839  4323 

28.96   , 

10 

i.  808  4389 

28.77 

1.818  8056 

28.82 

1.829  1931 

28.89 

1.839  6060 

28.96 

11 

.808  6116 

28.77 

.818  9785 

28.82 

.829  3665 

28.89 

.839  7798 

28.97 

12 

.808  7842 

28.77 

.819  1515 

28.83 

.829  5398 

28.89 

.839  9536 

28.97 

13 

.808  9568 

28.77 

.819  3244 

28.83 

.829  7131 

28.89 

.8^0  1274 

28.97 

14 

.809  1295 

28.77 

.819  4974 

28.83 

.829  8865 

28.89 

.840  3012 

28.97 

15 

1.809  3021 

28.78 

1.819  6704 

28.83 

1.830  0599 

28.89 

1.840  4751 

28.97 

16 

.809  4748 

28.78 

.819  8433 

28.83 

.830  2332 

28.89 

.840  6489 

28.97 

17 

.809  6474 

28.78 

.820  0163 

28.83 

.830  4066 

28.90 

.840  8227 

28.97   , 

18 

.809  8201 

28.78 

.820  1893 

28.83 

.830  5800 

28.90 

.840  9966 

28.97 

19 

.809  9928 

28.78 

.820  3623 

28.83 

•83°  7533 

28.90 

.841  1704 

28.98 

20 

1.810  1655 

28.78 

1.820  5353 

28.83 

1.830  9267 

28.90 

1.841  3443 

28.98 

21 

.810  3381 

28.78 

.820  7083 

28.83 

.831  looi 

28.90 

.841  5182 

28.98 

22 

.810  5108 

28.78 

.820  8813 

28.84 

•83!  2735 

28.90 

.841  6921 

28.98 

23 

.810  6835 

28.78 

.821  0543 

28.84 

.831  4470 

28.90 

.841  8659 

28.98 

24 

.810  8562 

28.78 

.821  2273 

28.84 

.831  6204 

28.90 

.842  0398 

28.98 

25 

1.811  0289 

28.78 

1.821  4003 

28.84 

I-831  7938 

28.91 

1.842  2138 

28.98 

26 

.811  2016 

28.78 

.821  5734 

28.84 

.831  9672 

28.91 

.842  3877 

28.99 

27 

.811  3743 

28.78 

.821  7464 

28.84 

.832  1407 

28.91 

842  5616 

28.99 

28 

.811  5470 

28.79 

.821  9194 

28.84 

.832  3141 

28.91 

•84*  7355 

28.99 

29 

.811  7197 

28.79 

.822  0925 

28.84 

.832  4876 

28.91 

.842  9095 

28.99 

30 

1.811  8924 

28.79 

1.822  2656 

28.84 

1.832  6611 

28.91 

1.843  0834 

28.99 

31 

.812  0652 

28.79 

.822  4386 

28.84 

.832  8345 

28.91 

.843  2574 

28.99 

32 

.812  2379 

28.79 

.822  6117 

28.85 

.833  0080 

28.92 

•843  43*3 

29.00 

33 

.812  4106 

28.79 

.822  7848 

28.85 

.833  1815 

28.92 

.843  6053 

29.00 

34 

.812  5834 

28.79 

.822  9578 

28.85 

•833  355° 

28.92 

•843  7793 

29.00 

35 

i.  812  7561 

28.79 

1.823  1309 

28.85 

1.833  5285 

28.92 

i-843  9533 

29.00 

36 

.812  9289 

28.79 

.823  3040 

28.85 

.833  7020 

28.92 

.844  1273 

29.00 

37 

.813  1016 

28.79 

.823  4771 

28.85 

•833  8755 

28.92 

.844  3013 

29.00 

38 

.813  2744 

28.79 

.823  6502 

28.85 

.834  0491 

28.92 

•844  4753 

29.00 

39 

.813  4472 

28.79 

.823  8233 

28.85 

.834  2226 

28.92 

.844  6494 

29.01 

40 

1.813  6199 

28.80 

1.823  9965 

28.85 

1.834  3961 

28.92 

1.844  8234 

29.01 

41 

.813  7927 

28.80 

.824  1696 

28.85 

.834  5697 

28.93 

.844  9974 

29.01 

42 
43 

•8i3  9655 
.814  1383 

28.80 
28.80 

.824  3427 
.824  5159 

28.86 
28.86 

•834  7432 
.834  9168 

2893 
28.93 

.845  1715 
.845  3456 

29.01 
29.01 

44 

.814  3111 

28.80 

.824  6890 

28.86 

.835  0904 

28.93 

.845  5196 

29.01   1 

45 

1.814  4839 

28.80 

1.824  8622 

28.86 

1.835  2640 

28.93 

1.845  6937 

29.01 

46 

.814  6567 

28.80 

•825  °353 

28.86 

.835  4376 

28.93 

.845  8678 

29.02 

47 

.814  8295 

28.80 

.825  2085 

28.86 

.835  6112 

28.93 

.846  0419 

29.02 

48 

.815  0023 

28.80 

.825  3816 

28.86 

.835  7848 

28.93 

.846  2160 

29.02 

49 

.815  1751 

28.80 

.825  5548 

28.86 

.835  9584 

28.94 

.846  3901 

29.02 

50 

1.815  3479 

28.80 

1.825  7280 

28.86 

1.836  1320 

28.94 

1.846  5643 

29  C2 

51 

.815  5208 

28.81 

.825  9012 

28.87 

.836  3056 

28.94 

.846  7384 

2O  O2 

52 

.815  6936 

28.81 

.826  0744 

28.87 

.836  4792 

28.94 

.846  9125 

29.03 

53 

.815  8664 

28.81 

.826  2476 

28.87 

.836  6529 

28.94 

.847  0867 

29.03 

54 

.816  0393 

28.81 

.826  4208 

28.87 

.836  8265 

28.94 

.847  2609 

29.03 

55 

1.816  2121 

28.81 

1.826  5940 

28.87 

1.837  0002 

28.94 

1.847  4350 

29.03 

56 

.816  3850 

28.81 

.826  7673 

28.87 

.837  1739 

28.95 

.847  6092 

29.03 

57 

58 

.816  5578 
.816  7307 

28.81 
28.81 

.826  9405 
.827  1137 

28.87 
28.87 

•837  3475 
.837  5212 

28.95 
28.95 

.847  7834 
.847  9576 

29.03 
29.03 

59 

.816  9036 

28.81 

.827  2870 

28.87 

.837  6949 

28.95 

.848  1318 

29.04 

60 

1.817  0765 

a8.8i 

1.827  4602 

28.88 

1.837  8686 

28.95 

1.848  3060 

29.04 

584 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit 


76° 

77° 

78° 

79° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0' 

1.848  3060 

29.04 

1.858  7769 

29.14 

1.869  2857 

29.25 

1.879  8369 

29-37 

1 

.848  4803 

29.04 

•858  9517 

29.14 

.869  4612 

29.25 

.880  0131 

29.37 

2 

.848  6545 

29.04 

.859  1266 

29.14 

.869  6367 

29.25 

.880  1894 

29.38 

3 
4 

.848  8287 
.849  0030 

29.04 
29.04 

•859  3OI4 
•859  4763 

29.14 
29.15 

.869  8122 
.869  9878 

29.25 
29.26 

.880  3656 
.880  5419 

29.38 
29.38 

5 

1-849  '773 

29.04 

1.859  6512 

29.15 

1.870  1633 

29.26 

1.  880  7182 

29.38 

6 

•849  35*5 

29.05 

.859  8260 

29.15 

.870  3389 

29.26 

.88c  8945 

29.38 

7 
8 

.849  5258 
.849  7001 

29.05 
29.05 

.860  0009 
.860  1758 

29-I5 
29.15 

.870  5144 
.870  6900 

29.26 
29.26 

.881  0708 
.881  2471 

29-39   1 
29.39   1 

9 

.849  8744 

29.05 

.860  3507 

29.15 

.870  8656 

29.26 

.881  4235 

29-39 

10 

1.850  0487 

29.05 

i.  860  5256 

29.15 

1.871  0412 

29.27 

1.881  5998 

29.39 

11 

.850  2231 

29.05 

.860  7006 

29.16 

.871  2168 

29.27 

.881  7762 

29-39 

12 

.850  3974 

29.06 

.860  8755 

29.16 

.871  3924 

29.27 

.881  9526 

29.40 

13 

.850  5717 

29.06 

.861  0505 

29.16 

.871  5681 

29.27 

.882  1290 

29.40 

14 

.850  7461 

29.06 

.861  2254 

29.16 

.871  7437 

29.28 

.882  3054 

29.40 

15 

1.850  9204 

29.06 

1.861  4004 

29.16 

1.871  9194 

29.28 

1.882  4818 

29.40 

16 

.851  0948 

29.06 

.861  5754 

29.16 

.872  0950 

29.28 

.882  6582 

29.41 

17 

.851  2692 

29.06 

.861  7504 

29.17 

.872  2707 

29.28 

.882  8347 

29.41 

18 

.851  4436 

29.07 

.861  9254 

29.17 

.872  4464 

29.28 

.883  0112 

29.41 

19 

.851  6180 

29.07 

.862  1004 

29.17 

.872  6221 

29.29 

.883  1876 

29.41 

20 

1.851  7924 

29.07 

1.862  2754 

29.17 

1.872  7979 

29.29 

1.883  3641 

29.42 

21 

.851  9668 

29.07 

.862  4505 

29.17 

.872  9736 

29.29 

.883  5406 

29.42 

22 

.852  1412 

29.07 

.862  6255 

29.18 

•873  "493 

29.29 

.883  7171 

29.42 

23 

.852  3157 

29.07 

.862  8006 

29.18 

.873  3251 

29.29 

.883  8937 

29.42 

24 

.852  4901 

29.07 

.862  9756 

29.18 

.873  5008 

29.30 

.884  0702 

29.42   , 

25 

1.852  6646 

29.08 

1.863  1507 

29.18 

1.873  6766 

29.30 

1.884  2468 

29-43 

26 

.852  8391 

29.08 

.863  3258 

29.18 

.873  8524 

29.30 

.884  4233 

29.43 

27 

•853  OI35 

29.08 

.863  5009 

29.18 

.874  0282 

29.30 

.884  5999 

29.43 

28 

.853  1880 

29.08 

.863  6760 

29.19 

.874  2041 

29.30 

-884  7765 

29-43 

29 

.853  3625 

29.08 

.863  8512 

29.19 

•874  3799 

29.31 

•884  9531 

29.44 

30 

I-853  537° 

29.09 

1.864  0263 

29.19 

1-874  5557 

29.31 

1.885  1297 

29.44 

31 

•853  7^5 

29.09 

.864  2015 

29.19 

.874  7316 

29.31 

.885  3064 

29.44 

32 

.853  8861 

29.09 

.864  3766 

29.19 

.874  9074 

29.31 

.885  4830 

29.44 

33 

.854  0606 

29.09 

.864  5518 

29.20 

.875  0833 

29.31 

.885  6597 

29.45 

34 

.854  2351 

29.09 

.864  7270 

29.20 

-875  2592 

29.32 

.885  8364 

29.45 

35 

1.854  4097 

29.09 

1.864  9022 

29.20 

1-875  435^ 

29.32 

1.886  0131 

29.45 

36 

•854  5843 

29.10 

.865  0774 

29.20 

.875  6m 

29.32 

.886  1898 

29.45   ' 

37 

.854  7588 

29.10 

.865  2526 

29.20 

.875  7870 

29.32 

.886  3665 

29.45 

38 

•854  9334 

29.10 

.865  4278 

29.20 

.875  9629 

29.32 

.886  5432 

29.46 

39 

.855  1080 

29.10 

.865  6030 

29.21 

.876  1389 

29.33 

.886  7200 

29.46 

40 
41 

1.855  2826 
.855  4572 

29.10 
29.10 

1.865  7783 
.865  9536 

29.21 
29.21 

1.876  3148 
.876  4908 

29-33 
29.33 

1.886  8967 
.887  0735 

29.46 
29.46 

42 

.855  6319 

29.11 

.866  1288 

29.21 

.876  6668 

29-33 

.887  2503 

29.47 

43 

.855  8065 

29.11 

.866  3041 

29.21 

.876  8428 

29-33 

.887  4271 

29.47 

44 

.855  9811 

29.11 

.866  4794 

29.22 

.877  0188 

29-34 

.887  6039 

29.47 

45 

1.856  1558 

29.11 

1.866  6547 

29.22 

1.877  1949 

29.34 

1.887  7807 

29.47 

46 

.856  3305 

29.11 

.866  8301 

29.22 

.877  3709 

29-34 

•887  9576 

29.48 

47 

.856  5052 

29.11 

.867  0054 

29.22 

.877  5470 

29-34 

.888  1344 

29.48 

48 

.856  6799 

29.12 

.867  1807 

29.22 

.877  7230 

29-34 

.888  3113 

29.48   I 

49 

.856  8546 

29.12 

.867  3561 

29.23 

.877  8991 

29-35 

.888  4882 

29.48 

50 

1.857  0293 

29.12 

1.867  5314 

29.23 

1.878  0752 

29.35 

1.888  6651 

29.48 

51 

.857  2040 

29.12 

.867  7068 

29.23 

.878  2513 

29.35 

.888  8420 

29-49 

52 

•857  3787 

29.12 

.867  8822 

29.23 

.878  4275 

29-35 

.889  0189 

29.49 

53 

•857  5534 

29.12 

.868  0576 

29.23 

.878  6036 

29-35 

.889  1959 

29.49 

54 

.857  7282 

29.13 

.868  2330 

29.24 

-878  7797 

29.36 

.889  3728 

29-49 

55 
56 

1.857  9030 
•858  0777 

29.13 
29.13 

1.868  4084 
.868  5839 

29.24 
29.24 

1.878  9559 
•879  1321 

29.36 
29.36 

1.889  5498 
.889  7268 

29-49 
29.50 

57 

.858  2525 

29.13 

.868  7593 

29.24 

.879  3082 

29.36 

.889  8038 

29.50 

58 

•858  4273 

29.13 

.868  9348 

29.24 

.879  4844 

29.36 

.890  0808 

29.50 

59 

.858  6021 

29.13 

.869  1102 

29.25 

.879  6606 

29-37 

.890  2578 

29.51 

60 

1.858  7769 

29.14 

1.869  2857 

29.25 

1.879  8369 

29-37 

1.890  4349 

2<M' 

585 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  I^erihelion  in  a  Parabolic  Orbit. 


V. 

80° 

81° 

82° 

83°    , 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

O' 

1.890  4349 

29.51 

1.901  0841 

29.66 

1.911  7893 

29.82 

1.922  5548 

29.99 

1 

.890  6119 

2951 

.901  2621 

29.66 

.911  9682 

29.82 

.922  7347 

29.99 

2 
3 

.890  7890 
.890  9661 

29-51 
29.51 

.901  4400 
.901  6180 

29.66 
29.66 

.912  1471 
.912  3261 

29.82 
29.83 

.922  9147 
.923  0947 

30.00 
30.00 

4 

.891  1432 

29.52 

.901  7960 

29.67 

.912  5050 

29.83 

.923  2747 

30.00 

5 

1.891  3203 

29.52 

1.901  9740 

29.67 

1.912  6840 

29.83 

1.923  4548 

30.01 

6 

.891  4974 

29.52 

.902  1521 

29.67 

.912  8630 

29.84 

.923  6348 

30.01 

7 

.891  6745 

29.52 

.902  3301 

29.67 

.913  0420 

29.84 

.923  8149 

30.01 

8 

.891  8517 

29-53 

.902  5082 

29.68 

.913  2211 

29.84 

.923  9950 

30.02 

9 

.892  0289 

29-53 

.902  6862 

29.68 

.913  4001 

29.84 

.924  1  75I 

30.02 

10 

1.892  2061 

29.53 

1.902  8643 

29.68 

I.9I3  5792 

29.85 

1.924  3552 

30.02 

11 

.892  3833 

29-53 

.903  0424 

29.69 

•913  7583 

29.85 

-924  5354 

30.03 

12 

.892  5605 

29.54 

.903  2105 

29.69 

•9i3  9374 

29.85 

.924  7155 

30.03 

13 

.892  7377 

29.54 

.903  3987 

29.69 

.914  1165 

29.85 

.924  8957 

30.03 

14 

.892  9149 

29.54 

.903  5768 

29.69 

.914  2956 

29.86 

•925  0759 

30.03 

15 

1.893  0922 

29.54 

i-9°3  755° 

29.70 

1.914  4748 

29.86 

1.925  2561 

30.04 

16 

.893  2695 

29-55 

.903  9332 

29.70 

.914  6540 

29.86 

.925  4364 

30.04 

17 

18 

.893  4467 
.893  6240 

29-55 
29-55 

.904  IIIA 
.904  2896 

29.70 
29.70 

-9H  8331 
.915  0124 

29.87 
29.87 

.925  6166 
.925  7969 

30.04 
30.05 

19 

.893  8013 

29-55 

.904  4678 

29.71 

.915  1916 

29.87 

.925  9772 

30.05 

2O 

1.893  9787 

29.56 

1.904  6461 

29.71 

1.915  3708 

29-87 

1.926  1575 

30.05 

21 

.894  1560 

29.56 

.904  8243 

29.71 

.915  5501 

29.88 

.926  3378 

30.06 

22 

•894  3334 

29.56 

.905  0026 

29.71 

.915  7294 

29.88 

.926  5182 

30.06 

23 

.894  5108 

29.56 

.905  1809 

29.72 

.915  9087 

29.88 

.926  6986 

30.06 

24 

.894  6882 

29.57 

.905  3592 

29.72 

.916  0880 

29.89 

.926  8789 

30.07 

25 

1.894  8656 

29.57 

i-9°5  5376 

29.72 

1.916  2673 

29.89 

1.927  0591 

30.07 

26 

.895  0430 

29.57 

•9°5  7159 

29-73 

.916  4466 

29.89 

.927  2398 

30.07 

27 

.895  2204 

29-57 

.905  8943 

29-73 

.916  6260 

29.90 

.927  4202 

30.08 

28 

•895  3979 

29.58 

.906  0726 

29-73 

.916  8054 

29.90 

.927  6007 

30.08 

29 

•895  5753 

29-58 

.906  2510 

29-73 

.916  9848 

29.90 

.927  7811 

30.08 

30 

1.895  7528 

29.58 

1.906  4294 

29-74 

1.917  1642 

29.90 

1.927  9616 

30.08 

31 

.895  9303 

29.58 

.906  6079 

29.74 

.917  3436 

29.91 

.928  1422 

30.09 

32 

.896  1078 

29.59 

.906  7863 

29.74 

.917  5231 

29.91 

.928  3227 

30.09 

33 

.896  2854 

29.59 

.906  9648 

29-74 

.917  7025 

29-9I 

.928  5032 

30.09 

34 

.896  4628 

29.59 

.907  1432 

29.75 

.917  8820 

29.92 

.928  6838 

30.10 

35 

1.896  6404 

29.59 

1.907  3217 

29.75 

1.918  0615 

29.92 

1.928  8644 

30.10 

36 

.896  8180 

29.60 

.907  5002 

29.75 

.918  2410 

29.92 

.929  0450 

30.10 

37 

.896  9955 

29.60 

.907  6787 

29.75 

.918  4206 

29.92 

.929  2256 

30.11 

38 

.897  1732 

29.60 

.907  8573 

29.76 

.918  6001 

29.93 

.929  4063 

30.11 

39 

.897  3508 

29.60 

.908  0358 

29.76 

.918  7797 

2993 

.929  5869 

30.11 

40 

1.897  5284 

29.61 

1.908  2144 

29.76 

1.918  9593 

29-93 

1.929  7676 

30.12 

41 

.897  7060 

29.61 

.908  3930 

29.77 

.919  1389 

29.94 

.929  9483 

30.12 

42 

.897  8837 

29.61 

.908  5716 

29.77 

.919  3185 

29.94 

.930  1291 

30.12 

43 

.898  0614 

29.61 

.908  7502 

29.77 

.919  4982 

29.94 

.930  3098 

30.13 

44 

.898  2390 

29.62 

.908  9288 

29.77 

.919  6778 

29-94 

•93°  4906 

30.13 

45 

1.898  4168 

29.62 

1.909  1075 

29.78 

1.919  8575 

29.95 

1.930  6713 

30.13 

46 

.898  5945 

29.62 

.909  2862 

29.78 

.920  0372 

29.95 

.930  8521 

30.13 

47 

.898  7722 

29.62 

.909  4648 

29.78 

.920  2169 

29-95 

.931  0330 

30.14 

48 

.898  9500 

29.63 

.909  6436 

29.78 

.920  3966 

29.96 

.931  2138 

30.14 

49 

.899  1277 

29.63 

.909  8223 

29.79 

.920  5764 

29.96 

.931  3946 

30.14 

50 

1.899  3055 

29.63 

1.910  ooio 

29.79 

1.920  7561 

29.96 

I-931  5755 

30.15 

51 

•899  4833 

29.63 

.910  1798 

29.79 

.920  9359 

29.97 

.931  7564 

30.15 

52 

.899  6611 

29.64 

.910  3585 

29.80 

.921  1157 

29.97 

•931  9373 

30.15 

53 

.899  8389 

29.64 

•9i°  5373 

29.80 

.921  2956 

29.97 

.932  1183 

30.16 

54 

.900  0168 

29.64 

.910  7161 

29.80 

.921  4754 

29.98 

.932  2992 

30.16 

55 

1.900  1946 

29.64 

1.910  8949 

29.80 

1.921  6552 

29.98 

1.932  4802 

30.16 

56 

.900  3725 

29.65 

.911  0738 

29.81 

.921  8351 

29.98 

.932  6612 

30.17 

57 

.900  5504 

29.65 

.911  2526 

29.81 

.922  0150 

29.98 

.932  8422 

30.17 

58 

.900  7283 

.911  4315 

29.81 

.922  1949 

29.99 

.933  0232 

30.17 

59 

.900  9062 

29.66 

.911  6104 

29.82 

.922  3748 

29.99 

.933  2043 

30.18 

60 

1.901  0841 

29.66 

1.911  7893 

29.82 

1.922  5548 

29.99 

1-933  3853 

30.18 

586 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

84° 

85° 

86° 

87° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  I". 

logM. 

Diff.  1". 

0' 

1 

1-933  3853 
•933  5664 

30.18 
30.18 

1.944  2856 
•944  4678 

30.38 
30.38 

1.955   2602 
•955  4438 

30.59 

30.60 

1.966   3140 
.966  4990 

30.82 
30.82 

2 

•933  7475 

30.19 

.944  6502 

3°-39 

•955  6274 

30.60 

.966   6839 

30.83 

3 

•933  9287 

30.19 

•944  8325 

3°-39 

.955  8110 

30.60 

.966   8689 

30-83 

4 

.934  1098 

30.19 

•945  OI48 

3°-39 

•955  9946 

30.61 

.967  0539 

30.84 

5 

1.934  2910 

30.20 

1.945  *972 

30.40 

1.956  1783 

30.61 

1.967   2389 

30.84 

6 

•934  4722 

30.20 

•945   3796 

30.40 

.956  3619 

30.61 

.967  4240 

30.84 

7 

•934  6533 

30.20 

•945  56zo 

30.40 

.956  5456 

30.62 

.967   6090 

30.85 

8 

•934  8346 

30.21 

•945  7444 

30.41 

.956  7294 

30.62 

.967   7941 

30.85 

9 

•935  0158 

30.21 

.945  9269 

30.41 

.956  9131 

30.63 

967   9792 

30.85 

10 

J-935  J97i 

30.21 

1.946  1094 

30.41 

1.957  0969 

30.63 

1.968    1644 

30.86 

11 

•935  3784 

30.22 

.946  2919 

30.42 

•957  ^807 

30.63 

.968    3496 

30.86 

12 

•935  5597 

30.22 

.946  4744 

30.42 

•957  4645 

30.64 

.968  5347 

30.87 

13 

•935  7410 

30.22 

.946  6569 

30.42 

•957  6483 

30.64 

.968  7200 

30.87 

14 

•935  9223 

30.22 

.946  8395 

3°-43 

•957  8322 

30.64 

.968  9052 

30.87 

15 

1.936  1037 

30.23 

1.947  0221 

3°-43 

1.958  0160 

30.65 

1.969  0905 

30.88 

16 
17 

.936  2851 
.936  4665 

30.23 
30.23 

.947  2047 
•947  3873 

3°-44 
3°-44 

•958  1999 
.958  3839 

30.65 
30.66 

.969  2757 
.969  4610 

30.88 
30.89 

18 

.936  6479 

30.24 

•947  5699 

3°-44 

•958  5678 

30.66 

.969  6464 

30.89 

19 

.936  8293 

30.24 

•947  75  *6 

3°-45 

.958  7518 

30.66 

.969  8317 

30.89 

20 

1.937  0108 

30.24 

'•947  9353 

3°-45 

1.958  9358 

30.67 

1.970  0171 

30.90 

21 

.937  1922 

30.25 

.948  1  1  80 

3°-45 

•959   "98 

30.67 

.970  2025 

30.90 

22 

•937  3737 

30.25 

.948  3007 

30.46 

•959  3°38 

30.67 

.970  3879 

30.91 

23 

•937  5553 

30.25 

•948  4834 

30.46 

•959  4879 

30.68 

.970  5734 

30.91 

24 

•937  7368 

30.26 

.948  6662 

30.46 

.959  6720 

30.68 

.970  7589 

30.91 

25 

1.937  9184 

30.26 

1.948  8490 

3°-47 

1.959  8561 

30.69 

1.970  9443 

30.92 

26 

.938  0999 

30.26 

•949  °3l8 

3°-47 

.960  0402 

30.69 

.971   1299 

30.92 

27 

.938  2815 

30.27 

.949  2146 

30.47 

.960  2243 

30.69 

•971   3'54 

30-93 

28 

.938  4632 

30.27 

•949  3975 

30.48 

.960  4085 

30.70 

.971  5010 

3°-93 

29 

.938  6448 

30.27 

•949  5804 

30.48 

.960  5927 

30.70 

.971  6866 

30-93 

30 

1.938  8264 

30.28 

J-949  7633 

30.48 

1.960  7769 

30.70 

1.971   8722 

30.94 

31 

.939  0081 

30.28 

-949  9462 

3°-49 

.960  9612 

30.71 

.972  0578 

30.94 

32 

-939  1898 

30.28 

.950  1291 

3°-49 

.961   1454 

30.71 

.972  2435 

30.95 

33 

•939  37i5 

30.29 

.950  3121 

30.50 

.961   3297 

30.71 

.972  4292 

30-95 

34 

•939  5533 

30.29 

•95°  4951 

30.50 

.961  5140 

30.72 

.972  6149 

30-95 

35 

M39  735° 

30.29 

1.950  6781 

30.50 

1.961  6983 

30.72 

1.972  8006 

30.96 

36 

.939  9168 

30.30 

.950  8611 

30-51 

.961  8827 

30.73 

.972  9864 

30.96 

37 

.940  0986 

30.30 

•95  i  O44i 

3°-5  ' 

.962  0671 

30-73 

•973  17^2 

30-97 

38 

.940  2804 

30.30 

.951  2272 

30.51 

.962  2515 

30-73 

•973  358o 

30-97 

39 

.940  4623 

30.31 

.951  4103 

30.52 

.962  4359 

3°-74 

•973  5438 

3°-97 

40 

1.940  6441 

30.31 

I-951   5934 

30.52 

1.962  6203 

30-74 

1.973  7297 

30.98 

41 

.940  8260 

30.31 

.951  7766 

30.52 

.962  8048 

3°-75 

•973  9*56 

30.98 

42 

.941  0079 

30.32 

•951  9597 

3°-53 

.962  9893 

30.75 

•974  1015 

30.99 

43 

.941   1898 

30.32 

.952  1429 

3°-53 

.963  1738 

30.75 

•974  2874 

3°-99 

44 

.941   3717 

30.3* 

.952  3261 

30-53 

.963  3583 

30.76 

•974  4734 

30.99 

45 

I-94I  5537 

30.33 

1.952  5093 

3°-54 

1.963  5429 

30.76 

1.974  6593 

31.00 

46 

•94i   7357 

3°-33 

.952  6925 

3°-54 

.963  7275 

3°-77 

•974  8454 

31.00 

47 

.941  9177 

3°-34 

.952  8758 

30-55 

.963  9121 

30.77 

.975  0314 

31.01 

48 

.942  0997 

3°-34 

•953  0591 

30-55 

.964  0967 

3°-77 

•975  2I74 

31.01 

49 

.942  2817 

3°-34 

•953  *4*4 

3°-55 

.964  2814 

30-78 

•9-^S  4°35 

31.01 

50 

1.942  4638 

3°-35 

1.953  4*57 

30.56 

1.964  4660 

30.78 

1.975  5896 

31.02 

51 

.942  6459 

3°-35 

•953  6o9! 

30.56 

.964  6507 

30.78 

•975  7757 

31.02 

52 

.942  8280 

3°-35 

•953  79*4 

30.56 

.964  8354 

30-79 

•975  96l9 

31.03 

53 

.943  oioi 

30.36 

•953  9758 

30-57 

.965    0202 

30.79 

.976   1481 

3«-°3 

54 

•943  '9*3 

30.36 

•954  159* 

30-57 

.965    2050 

30.80 

•976  3343 

31.04 

55 
56 
57 

1-943  3744 
•943  5566 
•943  7388 

30.36 
30-37 
3°-37 

J-954  34^7 
•954  526z 
•954  7°9° 

30-57 
30.58 

30-58 

1.965    3897 
.965    5746 

•965  7594 

30.80 
30.80 
30.81 

1.976  5205 
.976  7067 
.976  8930 

31.04 
31.04 
31.05 

58 
59 

•943  9^11 
•944  I033 

30-37 
30-38 

•954  8931 
-955  °766 

30-59 
30-59 

.965  9442 
.966   1291 

30.81 
30.81 

•977  0793 
.977  2656 

31.05 
31.06 

60 

1.944  2856 

30.38 

1.955  2602 

30-59 

1.966  3140 

30.82 

1.977  4520 

31.06 

587 


TABLE  VI 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Oibit. 


V. 

88° 

89° 

90° 

91° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

o 

1.977  4520 

31.06 

1.988  6789 

31.31 

2.0OO  0000 

31.58 

2.01  I  4203 

31-87 

1 

•977  6383 

31.06 

.988  8668 

31.32 

.000  1895 

31.59 

.on  6115 

31.87 

2 

.977  8247 

31.07 

.989  0548 

31-3* 

.000  3790 

31-59 

.011  8027 

31.88 

3 
4 

.978  0112 
.978  1976 

31.07 
31.08 

.989  2427 
.989  4307 

3*-33 
31-33 

.000  5686 
.000  7582 

31.60 

31.60 

.on  9940 

.012  1853 

31.88 
31.89 

5 

1.978  3841 

31.08 

1.989  6187 

3*-34 

2.000  9478 

31.60 

2.012  3766 

31.89 

6 

.978  5706 

31.08 

.989  8067 

3M4 

.001  1375 

31.61 

.012  5680 

31.89 

7 

.978  7571 

31.09 

.989  9948 

3'-34 

.001  3272 

31.61 

.012  7594 

31.90 

8 

.978  9436 

31.09 

.990  1829 

3'-35 

.001  5169 

31.62 

.012  9508 

31.90 

9 

.979  1302 

31.10 

.990  3710 

3'-35 

.001  7066 

31.62 

.013  1422 

31.91 

10 

1.979  3168 

31.10 

1.990  5591 

3'-36 

2.001  8963 

31.63 

2.013  3337 

31.91 

11 

•979  5034 

31.11 

.990  7473 

31.36 

.OO2  O§6l 

31.63 

.013  5252 

31.92 

12 

•979  6901 

31.11 

.990  9355 

3'-37 

.002  2759 

31.64 

.013  7167 

31.92 

13 

.979  8768 

31.11 

.991  1237 

31-37 

.002  4658 

31.64 

.013  9083 

31-93 

14 

.980  0635 

31.12 

.991  3119 

3'-38 

.002  6557 

31.65 

.014  0999 

3i  93 

15 

1.980  2502 

31.12 

1.991  5002 

3'-38 

2.002  8456 

3'-65 

2.014  29J5 

31.94 

16 

.980  4369 

3'-i3 

.991  6885 

3I-38 

.003  0355 

31.66 

.014  4831 

31.94 

17 

.980  6237 

31-I3 

.991  8768 

31.39 

.003  2254 

31.66 

.014  6748 

3'-95 

18 

.980  8105 

3**13 

.992  0651 

3'-39 

.003  4154 

31.67 

.014  8665 

3'-95 

19 

.980  9973 

S'-H 

•99*  *535 

31.40 

.003  6054 

31.67 

.015  0582 

31.96 

20 

1.981  1842 

S'-H 

1.992  4419 

31.40 

2.003  7955 

31.68 

2.015  2500 

31.96 

21 

.981  3710 

31-iS 

.992  6304 

31.41 

.003  9855 

31.68 

.015  4418 

31-97 

22 

.981  5579 

I1-1* 

.992  8188 

3J-41 

.004  1756 

31.68 

.015  6336 

3*-97 

23 

.981  7449 

31.16 

•993  °°73 

31.42 

.004  3658 

31.69 

.015  8255 

31.98 

24 

.981  9318 

31.16 

•993  *958 

31.42 

.004  5559 

31.69 

.016  0174 

31.98 

25 

1.982  1188 

31.16 

1.993  3843 

31.42 

2.004  7461 

31.70 

2.016  2093 

3'-99 

26 

.982  3058 

31.17 

•993  57^9 

3!-43 

.004  9363 

31.70 

.016  4012 

31.99 

27 

.982  4928 

31.17 

•993  7615 

3'-43 

.005  1265 

31.71 

.016  5932, 

32.00 

28 

.982  6798 

31.18 

•993  9501 

3'  -44 

.005  3168 

3'-7i 

.016  7852 

32.00 

29 

.982  8669 

31.18 

•994  '387 

31.44 

.005  5071 

31.72 

.016  9772 

32.01 

30 

1.983  0540 

31.18 

1.994  3274 

31-45 

2.005  6974 

31.72 

2.017  T*>93 

32.01 

31 

.983  2411 

31.19 

•994  5'61 

31-45 

.005  8878 

3I-73 

.017  3614 

32.02 

32 
33 

.983  4283 
.983  6155 

31.19 
31.20 

•994  7048 
•994  8936 

31.46 
31.46 

.006  0781 
.006  2685 

31-73 
3*-74 

-017  5535 
.017  7456 

32.02 
32.03 

34 

.983  8027 

31.20 

•995  °8i3 

31.46 

.006  4590 

3!-74 

.017  9378 

3*-°3 

35 

1.983  9899 

31.21 

1.995  2711 

3M7 

2.006  6494 

31-75 

2.018  1300 

32.04 

36 

.984  1772 

31.21 

.995  4600 

31-47 

.006  8399 

31-75 

.018  3223 

32.04 

37 

.984  3644 

31.22 

.995  6488 

31.48 

.007  0304 

31.76 

.018  5145 

32.05 

38 

.984  5517 

31.22 

•995  8377 

31.48 

.007  22  I  O 

31.76 

.018  7068 

32.05 

39 

.984  7391 

31.22 

.996  0266 

31.49 

.007  4116 

3«-77 

.018  8992 

32.06 

40 

1.984  9264 

31.23 

1.996  2155 

3J-49 

2.007  6022 

3'-77 

2.019  °9I5 

32.06 

41 

.985  1138 

31.23 

.996  4045 

3!-5° 

.007  7928 

3'-77 

.019  2839 

32.07 

42 

.985  3012 

31.24 

.996  5935 

3l-SQ 

.007  9835 

3I-78 

.019  4763 

32.07 

43 

.985  4886 

31.24 

.996  7825 

31.51 

.008  1742 

31.78 

.019  6688 

32.08 

44 

.985  6761 

31.24 

.996  9716 

3!'5i 

.008  3649 

31.79 

.019  8613 

32.08   : 

45 

1.985  8636 

31.25 

1.997  1606 

31.51 

2.008  5556 

31.79 

2.020  0538 

32.09 

46 

.986  0511 

3r-25 

•997  3497 

31.52 

.008  7464 

31.80 

.020  2463 

32.09 

47 

.986  2386 

31.26 

•997  5389 

3T-52 

.008  9372 

31.80 

.020  4389 

32.10 

48 

.986  4262 

31.26 

.997  7280 

3!-53 

.009  I28o 

31.81 

.O2O  6315 

32.10 

49 

.986  6138 

31.27 

•997  9J7* 

3'-53 

.009  3189 

31.81 

.020  8241 

32.11 

50 

1.986  8014 

31.27 

1.998  1064 

31-54 

2.009  5°98 

31.82 

2.021  Ol68 

32.11 

51 

.986  9890 

31.28 

.998  2956 

3'-54 

.009  7007 

31.82 

.021  2095 

32.12 

52 

.987  1767 

31.28 

.998  4849 

3'-55 

.009  8917 

3i-83 

.021  4022 

32.12 

53 

.987  3644 

31.28 

.998  6742 

3'-55 

.010  0826 

31.83 

.021  5949 

32.13 

54 

.987  5521 

31.29 

.998  8635 

3'-56 

.010  1736 

31.84 

.021  7877 

3*-»3 

55 

1.987  7398 

31.29 

1.999  0529 

3i-5f 

2.OIO  4647 

31.84 

2.021  9805 

32.14 

56 

.987  9276 

31.30 

.999  2422 

31.56 

.OIO  6557 

31.85 

.022  1734 

32.14 

57 

.988  1154 

31.30 

•999  43'6 

31-57 

.010  8468 

31-85 

.022  3662 

32.15 

58 

.988  3032 

31.31 

.999  6211 

31-57 

.on  0380 

31.86 

.022  5591 

J*.i| 

59 

.988  4911 

31.31 

•999  8l°5 

3'-58 

.on  2291 

31.86 

.022  7521 

32.16 

60 

1.988  6789 

S1^1 

2.000  0000 

31.58 

2.01  1  4103 

31.87 

2.O22  945O 

32.16 

-^J 

588 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

92° 

93° 

94° 

95° 

log  M. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

1  gM. 

Diff.  1". 

0' 

.022  9450 

32.16 

1-034  5797 

32.48 

2.046  3296 

32.80 

2.058  2005 

33-15 

1 

.023  1380 

32.17 

•°34  7745 

32.48 

.046  5264 

32.81 

.058  3994 

33-15 

2 

.023  33II 

32.17 

.034  9694 

32-49 

.046  7233 

32.82 

.058  5983 

33-i6 

3 

.023  5241 

32.18 

.035  1644 

32.49 

.046  9202 

32.82 

.058  7973 

33.16 

4 

.023  7172 

32.18 

•035  3593 

32.50 

.047  1172 

32.83 

.058  9963 

31-17 

5 

2.023  9103 

32.19 

•035  5543 

32.50 

2.047  3141 

32.83 

2.059  1953 

33-i8 

6 

.024  1035 

32.19 

.035  7494 

3^-51 

.047  5111 

32-84 

•°59  3944 

33-i8 

7 

.024  2967 

32.20 

•035  9444 

32-51 

.047  7082 

32.84 

•059  5935 

33-19 

8 

.024  4899 

32.20 

.036  1395 

32.52 

.047  9053 

32.85 

.059  7927 

33-19 

9 

.024  6831 

32.21 

.036  3347 

32.52 

.048  1024 

32.85 

.059  9919 

33-20 

10 

2.024  8764 

32.21 

2.036  5298 

32.53 

2.048  2995 

32.86 

2.060  1911 

33.21 

11 

.025  0697 

32.22 

.036  7250 

32.53 

.048  4967 

32.87 

.060  3904 

33-21 

12 

.025  2630 

32.22 

.036  9202 

32.54 

.048  6939 

32-87 

.060  5897 

33.22 

13 

.025  4564 

32.23 

.037  1155 

32-54 

.048  8912 

32.88 

.060  7890 

33-22 

14 

.025  6498 

32.23 

.037  3108 

32.55 

.049  0884 

32.88 

.060  9884 

33-23 

15 

2.025  8432 

32.24 

2.037  5061 

32.55 

2.049  2857 

32.89 

2.061  1878 

33-24 

16 

.026  0367 

32.24 

.037  7015 

32.56 

.049  4831 

32.89 

.061  3872 

33-24 

17 

.026  2301 

32.25 

.037  8969 

32.57 

.049  6805 

32.90 

.061  5867 

33-25 

18 

.026  4236 

32.26 

.038  0923 

32.57 

.049  8879 

32.90 

.061  7862 

33-25 

19 

.026  6172 

32.26 

.038  2877 

32-58 

.050  0753 

32.91 

.061  9857 

33.26 

20 

2.026  8lo8 

32.27 

2.038  4832 

32.58 

2.050  2728 

32.92 

2.062  1853 

33-27 

21 

.027  0044 

32.27 

.038  6787 

32.59 

.050  4703 

32-92 

.062  3849 

33-27 

22 

.027  1980 

32.28 

.038  8743 

32.59 

.050  6679 

32.93 

.062  5846 

33-28 

23 

.027  3917 

32.28 

.039  0699 

32.60 

.050  8655 

32.93 

.062  7842 

33.28 

24 

.027  5854 

32.29 

.039  2655 

32.61 

.051  0631 

32.94 

.062  9840 

33-29 

25 

2.027  7791 

32.29 

2.039  4611 

32.61 

2.051  2608 

32.95 

2.063  1837 

33-3° 

26 
27 

.027  9729 
.028  1667 

323° 
32.3° 

.039  6568 
.039  8525 

32.62 
32.62 

.051  4585 
.051  6562 

32.95 
32.96 

.063  3835 
.063  5833 

33-3° 
33-31 

28 

.028  3605 

32-31 

.040  0482 

32.63 

.051  8539 

32.96 

.063  7832 

33-31 

29 

.028  5544 

32.31 

.040  2440 

32.63 

.052  0517 

32.97 

.063  9831 

33.32 

30 

2.028  7483 

32-3* 

2.040  4399 

32.64 

2.052  2496 

32.97 

2.064  l83i 

33-33 

31 

.028  9422 

32.32 

.040  6357 

32.64 

.052  4474 

32-98 

.064  3830 

33-33 

32 

.029  1361 

32-33 

.040  8316 

32.65 

.052  6453 

32.98 

.064  5830 

33-34 

33 

.029  3301 

32-33 

.041  0275 

32.65 

.052  8432 

32.99 

.064  7831 

33-34 

34 

.029  5241 

32.34 

.041  2234 

32.66 

.053  0412 

33.00 

.064  9832 

33-35 

35 

2.029  7182 

32.34 

2.041  4194 

32.67 

2053  2392 

33.00 

2.065  l833 

33-36 

36 
37 

.029  9123 
.030  1064 

32.35 
32.35 

.041  6154 
.041  8114 

32.67 
32.68 

•°53  4372 
.053  6353 

33-oi 
33.01 

.065  3834 
.065  5836 

33-36 
33-37 

38 

.030  3005 

32.36 

.042  0075 

32.68 

•°53  8334 

33.02 

.065  7839 

33-37 

39 

.030  4947 

32.36 

.042  2036 

32.69 

.054  0315 

33-°3 

.065  9841 

33-38 

40 

2.030  6889 

32-37 

2.042  3998 

32.69 

2.054  2297 

33-03 

2.066  1844 

33-39 

41 

.030  8831 

32-37 

.042  5960 

32.70 

.054  4279 

33-°4 

.066  3847 

33-39 

42 

.031  0774 

32.38 

.042  7922 

32.70 

.054  6262 

33-°4 

.066  5851 

33-4° 

43 

.031  2717 

32.39 

.042  9834 

32.71 

.054  8244 

33-05 

.066  7855 

33-40 

44 

.031  4660 

32.39 

.043  1847 

32.71 

.055  0227 

33-°5 

.066  9860 

33.41 

45 

2.031  6604 

32.40 

2.043  3810 

32.72 

2.055  221  1 

33.06 

2.067  1865 

33-42 

46 

.031  8548 

32.40 

•043  5773 

32.73 

.055  4195 

33.07 

.067  3870 

33-42 

47 

.032  0492 

32.41 

•043  7737 

32.73 

.055  6179 

33-07 

.067  5875 

33-43 

48 

.032  2437 

32.41 

.043  9701 

32.74 

.055  8l6l 

33.08 

.067  7881 

33-43 

49 

.032  4382 

32.42 

.044  1665 

32.74 

.056  0148 

33.08 

.067  9887 

33-44 

50 

2.032  6327 

32.42 

2.044  3630 

32-75 

2.056  2133 

33-°9 

2.068  1894 

33-45 

51 
52 
53 

.032  8272 
.033  0218 
.033  2164 

32.43 
32-43 
32-44 

•044  5595 
.044  7561 
.044  9526 

32-75 
32.76 
32.76 

.056  4119 
.056  6105 
.056  8091 

33.10 
33.10 
33-" 

.068  3901 
.068  5908 
.068  7916 

33-45 
33-46 
33-47 

54 

.033  4111 

32-44 

.045  1492 

32.77 

.057  0078 

33-" 

.068  9924 

33-47 

55 

2.033  6058 

32.45 

2.045  3459 

32.78 

2.057  2065 

33.12 

2.069  1933 

3348 

56 

.033  8005 

32-45 

.045  5426 

32.78 

.057  4052 

33.12 

.069  3942 

33-48 

57 

•°33  9952 

32.46 

•°45  7393 

32.79 

.057  6040 

33.13 

.069  5951 

33-49 

58 

.034  1900 

32-47 

.045  9360 

32-79 

.057  8028 

33-14 

.069  7960 

33.50 

59 

.034  3848 

32.47 

.046  1328 

32.80 

.058  00l6 

33-H 

.069  9970 

33.50 

60 

2.034  5797 

3^.48 

2.046  3296 

32.80 

2.058  2005 

33-*5 

2.070  1980 

33-51 

589 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

96° 

97° 

98° 

99° 

logic. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Dtff.  l". 

0' 

2.070  1980 

33-51 

2.082  3282 

33-88 

2.094  5971 

34.28 

2.107  0109 

34.69 

1 

.070  3991 

33-51 

.082  5316 

33-89 

.094  8028 

34.29 

.107  2190 

34-70 

2 

.070  6002 

33-52 

.082  7349 

33-9° 

.095  0085 

34.29 

.107  4272 

34.70 

3 

.070  8014 

33-53 

.082  9383 

33-90 

.095  2143 

34-3° 

.107  6355 

34-71 

4 

.071  0025 

33-53 

.083  1418 

33-91 

.095  4201 

34.31 

.107  8437 

34-72 

5 

2.071  2037 

33-54 

2-083  3453 

33-92 

2.095  6260 

34-31 

2.108  0521 

34-72 

6 

.071  4050 

33-54 

.083  5488 

33-92 

.095  8318 

34-32 

.108  2604 

34-73   i 

7 

.071  6063 

33-55 

.083  7523 

33-93 

.096  0378 

34.33 

.108  4689 

34-74 

8 

.071  8076 

33-56 

.083  9559 

33-94 

.096  2438 

34-33 

.108  6773 

34-75 

9 

.072  0090 

33.56 

.084  1596 

33-94 

.096  4498 

34-34 

.108  8858 

34-75 

10 

2.072  2104 

33-57 

2.084  3633 

33-95 

2.096  6558 

34-35 

2.109  0944 

34-76 

11 

.072  4118 

33.58 

.084  5670 

33-96 

.096  8619 

34-35 

.109  3029 

34-77 

12 

.072  6133 

33-58 

.084  7707 

33-96 

.097  0681 

34-36 

.109  5116 

34-77 

13 

-.072  8148 

33-59 

.084  9745 

33-97 

.097  2742 

34-37 

.109  7202 

34.78 

14 

.073  0163 

33  59 

.085  1783 

33.98 

.097  4804 

34-37 

.109  9289 

34-79 

15 

2.073  2179 

33.60 

2.085  3822 

33.98 

2.097  6867 

34.38 

2.IIO  1377 

34.80 

16 

.073  4195 

33.61 

.085  5861 

33-99 

.097  8930 

34-39 

.no  3465 

34.80 

17 

.073  6212 

33.61 

.085  7901 

33-99 

.098  0993 

34-39 

•"o  5553 

34.81 

18 

.073  8229 

33.62 

.085  9941 

34.00 

.098  3057 

34-4° 

.no  7642 

34-82 

19 

.074  0246 

33-63 

.086  1981 

34.01 

.098  5121 

34-41 

.no  9731 

34.82 

20 

2.074  2264 

33-63 

2.086  4021 

34.01 

2.098  7186 

34-41 

2.  ni  1821 

34.83 

21 

.074  4282 

33-64 

.086  6062 

34.02 

.098  9251 

3442 

.in  3911 

34.84 

22 

.074  6301 

33-64 

.086  8104 

34-03 

.099  1316 

34-43 

.in  6001 

34.85 

23 

.074  8320 

33-65 

.087  0146 

34-03 

.099  3382 

34-43 

.in  8092 

34-85 

24 

•°75  °339 

33.66 

.087  2188 

34-04 

•°99  5449 

34-44 

.HZ  0184 

34-86 

25 

2.075  2358 

33.66 

2.087  4231 

34-05 

2.099  7515 

34-45 

2.II2  2275 

34-87 

26 

.075  4378 

33-67 

.087  6274 

34.05 

.099  9582 

34-45 

.112  4368 

34-87 

27 

.075  6399 

33-67 

.087  8317 

34-o6 

.100  1650 

34-46 

.112  6460 

34-88 

28 

.075  8419 

33-68 

.088  0361 

34-07 

.100  3718 

34-47 

.112  8553 

34.89 

29 

.076  0440 

33-69 

.088  2405 

34.07 

.100  5786 

34.48 

.113  0647 

34-90 

30 

2.076  2462 

33.69 

2.088  4449 

34.08 

2.100  7855 

34-48 

2.II3  2741 

34.90 

31 
32 

.076  4484 
.076  6507 

33-7° 
33.71 

.088  6494 
.088  8540 

34.09 
34-09 

.100  9924 
.101  1993 

34-49 
34.50 

.113  4835 
.113  6930 

34-91 
34-92 

33 

.076  8529 

33-71 

.089  0586 

34.10 

.101  4063 

34.50 

.113  9025 

34-92 

34 

.077  0552 

33-72 

.089  2632 

34-n 

.101  6134 

34-51 

.114  II2I 

34-93 

35 

2.077  2575 

33-73 

2.089  4678 

34-" 

2.IOI  8204 

34-52 

2.114  3*!7 

34-94 

36 

.077  4599 

3J-73 

.089  6725 

34.12 

.102  0276 

34-52 

•"4  53J3 

34-95 

37 

.077  6623 

33-74 

.089  8772 

34.12 

.102  2347 

34-53 

.114  7410 

34-95 

38 

.077  8647 

33-74 

.090  0820 

34-13 

.102  4419 

34-54 

.114  9508 

34.96 

39 

.078  0672 

33-75 

.090  2868 

34.14 

.IO2  6492 

34-54 

.115  1605 

34-97 

40 

2.078  2697 

33-76 

2.090  4917 

34-15 

2.102  8564 

34-55 

2.115  3704 

34-97 

41 

.078  4723 

33-76 

.090  6966 

34-15 

.103  0638 

34-56 

.115  5802 

34-98 

42 

.078  6749 

33-77 

.090  9015 

34.16 

.103  2711 

34.56 

.115  7901 

34-99 

43 

.078  8775 

33-78 

.091  1065 

34-17 

.103  4785 

34-57 

.116  oooi 

35-0° 

44 

.079  0802 

33-78 

.091  3115 

34-17 

.103  6860 

34.58 

.Il6  2101 

35-oo 

45 

2.079  2829 

33-79 

2.091  5165 

34.18 

2.103  8935 

34-59 

2.116  4201 

35-oi 

46 

.079  4857 

33.80 

.091  7216 

34.19 

.  1  04  i  o  i  o 

34-59 

.116  6301 

35-02 

47 

.079  6885 

3380 

.091  9268 

34.19 

.104  3086 

34.60 

.116  8403 

35-02 

48 

.079  8913 

33.81 

.092  1319 

34-20 

.104  5162 

34.61 

.117  0505 

35-03 

49 

.080  0942 

33-8i 

.092  3371 

34-20 

.104  7239 

34.61 

.117  2607 

35-04 

50 

2.080  2971 

33-82 

2.092  5424 

34.21 

2.104  93J6 

34.62 

2.117  4710 

35-05 

51 

.080  5000 

3383 

.092  7477 

34-22 

.105  1393 

34-63 

.117  6813 

35-05 

1  52 

.080  7030 

33-83 

.092  9530 

34-22 

.105  3471 

34.63 

.117  8916 

35.06 

53 

.080  9060 

33-84 

.093  1584 

34-23 

-105  5549 

34-64 

.Il8  1020 

35-07 

54 

.081  1091 

33-85 

.093  3638 

34.24 

.105  7628 

34-65 

.Il8  3124 

35.08 

55 

2.081  3122 

33-85 

2.093  5692 

34.25 

2.105  97°7 

34-66 

2.118  5229 

35.08 

56 

.081  5153 

33-86 

.093  7747 

34.25 

.106  1786 

34-66 

.118  7334 

35-09 

57 

.081  7185 

33-87 

.093  9803 

34-26 

.106  3866 

34.67 

.118  9440 

35-10 

58 

.081  9217 

33-87 

.094  1858 

34-27 

.106  5947 

34-68 

,119  1546 

35-10 

59 

.082  1249 

33-88 

.094  3914 

34-27 

.106  8027 

34.68 

.119  3652 

35-" 

60 

2.082  3282 

33-88 

2.094  5971 

34.28 

2.107  0109 

34.69 

2.119  5759 

35-ia 

590 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

100° 

101° 

102° 

103° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff  I''. 

0 

2.119  5759 

35-12 

2.132  2989 

35-57 

2.145  *866 

36.03 

2.158  2460 

36.52 

1 

.119  7867 

35-13 

.132  5123 

35-57 

.145  4028 

36.04 

.158  4652 

36.53 

2 

.119  9974 

35-13 

.132  7258 

35.58 

.145  6191 

36.05 

.158  6844 

36.54 

3 

.120  2083 

35-H 

•132  9393 

35-59 

•H5  8354 

36.06 

.158  9036 

36.55 

4 

.120  4191 

35-15 

.133  1529 

35.60 

.146  0518 

36.07 

.159  I229 

36.55  ' 

5 

2.120  6301 

35.16 

2-133  3665 

35-6i 

2.146  2682 

36.07 

2.159  3423 

36.56 

6 

.120  8410 

35-16 

.133  5802 

35-6i 

.146  4847 

36.08 

.159  5617 

36-57 

7 

.121  0520 

35-17 

•!33  7939 

35-62 

.146  7012 

36.09 

.159  78ll 

36.58 

8 

.121  2630 

.134  0076 

35-63 

.146  9178 

36.10 

.160  0006 

36.59 

9 

.121  4741 

35-19 

.134  2214 

35-64 

.147  1344 

36.11 

.l6o  2202 

36.60 

10 

2.  121  6853 

35-19 

2.134  4352 

35-64 

2.147  3510 

36.11 

^.160  4398 

36.60 

11 

.121  8965 

35.20 

.134  6491 

35.65 

.147  5677 

36.12 

.l6o  6594 

36.61 

12 

.122  1077 

35-21 

.134  8631 

35-66 

.147  7845 

36.13 

.l6o  8791 

36.62 

13 

.122  3190 

.135  0770 

35-67 

.148  0013 

36.14 

,l6l  0989 

36.63 

14 

.122  5303 

35-22 

.135  2910 

35-67 

.148  2182 

36.15 

.l6l  3187 

36.64 

15 

2.122  7416 

35-23 

2.135  5051 

35-68 

2.148  4351 

36.15 

2.161  5385 

36.65 

16 

.122  9530 

35-24 

.135  7192 

35.69 

.148  6520 

36.16 

.161  7584 

36.65 

17 

.123  1644 

35-24 

•*35  9334 

35.70 

.148  8690 

36.17 

.161  9784 

36.66 

18 

.123  3759 

35-25 

.136  1476 

35.71 

.149  0861 

36.18 

.162  1984 

36.67 

19 

.123  5875 

35.26 

.136  3619 

35-71 

.149  3032 

36.19 

.162  4185 

36.68 

20 

2.123  7990 

35-27 

2.136  5762 

35-72 

2.149  5203 

36.19 

2.162  6386 

36.69 

21 

.124  0107 

35-27 

.136  7905 

35-73 

•H9  7375 

36.20 

.162  8587 

36.70 

22 

.124  2223 

35-28 

.137  0049 

35-74 

.149  9547 

36.21 

.163  0789 

36.70 

23 

.124  4340 

35-29 

.137  2193 

35-74 

.150  1720 

36.22 

.163  2992 

36.71 

24 

.124  6458 

35-3° 

•*37  4338 

35-75 

.150  3893 

36-23 

.163  5195 

36.72 

25 

2.124  8576 

35-3° 

2.137  6484 

35-76 

2.150  6067 

36.23 

2.163  7398 

36.73 

26 

.125  0694 

35-31 

.137  8630 

35-77 

.150  8242 

36-24 

.163  9602 

36.74 

27 

.125  2813 

35-32 

.138  0776 

35-77 

.151  0417 

36.25 

.164  1807 

36.74 

28 
29 

.125  4933 
.125  7052 

35-33 
35-33 

.138  2922 
.138  5070 

35-78 
35-79 

.151  2592 
.151  4768 

36.26 
36.27 

.164  4012 

.164  6218 

36.75 
36.76 

30 

2.125  9173 

35-34 

2.138  7217 

35.80 

2.151  6944 

36.28 

2.164  8424 

36.77 

31 

.126  1293 

35-35 

•138  9365 

.151  9121 

36.28 

.165  0630 

36.78 

32 

.126  3414 

35-35 

.139  1514 

35'8i 

.152  1298 

36.29 

.165  2837 

36.79 

33 

.126  5536 

35-36 

.139  3663 

35-82 

.152  3476 

36.30 

.165  5045 

36.80 

34 

.126  7658 

35  37 

.139  5813 

35-83 

.152  5654 

36.31 

.165  7253 

36.81 

35 
36 

2.126  9780 
.127  1903 

35-38 
35-39 

2.139  7963 
.140  0113 

35-84 
35.84 

2.152  7833 

.153  OO  I  2 

36.32 
36.32 

2.165  9462 
.166  1671 

36.81 

36.82 

37 

.127  4027 

35-39 

.140  2264 

35.85 

.153  2I92 

36.33 

.166  3881 

36-83 

38 

.127  6151 

35-4° 

.140  4415 

35-86 

•153  4372 

36.34 

.166  6091 

36.84 

39 

.127  8275 

35-41 

.140  6567 

35-87 

•153  6552 

36.35 

.166  8301 

36-85 

40 

2.128  0400 

35-42 

2.140  8720 

35-88 

2.153  8734 

36.35 

2.167  0513 

36.86 

41 
42 
43 
44 

.128  2525 
.128  4650 
.128  6776 
.128  8903 

35.42 
35-43 
35-44 
35-45 

.141  0873 
.141  3026 
.141  5180 
.141  7334 

35-88 
35.89 
35-9° 
35.91 

.154  0915 
.154  3097 
.154  5280 
.154  7463 

36.36 
36.37 
36.38 

.167  2724 
.167  4936 
.167  7149 

.167  9362 

36.87 
36.87 
36.88 
36.89 

45 
46 

47 
48 
19 

2.129  1030 
.129  3157 
.129  5285 
.129  7414 
.129  9542 

35-45 
35-46 
35-47 
3548 
35.48 

2.141  9489 
.142  1644 
.142  3799 
.142  5955 
.142  8112 

35.92 
35-93 
35-94 
35-95 

2.154  9647 
.155  1831 
.155  4015 
.155  6200 
.155  8386 

36.40 
36.41 
36-41 
36.42 
36.43 

2.168  1576 
.168  3790 
.168  6005 

.168  8220 
.169  0436 

36.90 

36.91 
36.92 

36.93 
36.93 

50 

2.130  1672 

35-49 

2.143  0269 

35-96 

2.156  0572 

36.44 

2.169  2652 

36.94 

51 
52 
53 
54 

.130  3801 

•13°  5931 
.130  8062 
.131  0193 

35-5° 
35-51 
35-5i 
35-52 

.^43  2427 
-143  4585 
-143  6743 
.143  8902 

35-96 
35-97 
35.98 
35-99 

.156  2759 
.156  4946 
.156  7133 
.156  9321 

36.45 

3646 
36.47 

.169  4869 
.169  7087 

.169  9304 
.170  1523 

36.96 
36.97 

36.98 

55 
56 
57 

2.131  2325 

•131  4457 
.131  6589 

35-53 
35-54 
35-54 

2.144  1062 
.144  3222 
.144  5382 

36.00 
36.00 
36.01 

2.157  i510 
.157  3699 
.157  5889 

36.48 

36.49 
36.50 

1.170  3742 
.170  5961 

.170  8181 

36.99 

36-99 
37.00 

58 
59 

.131  8722 
.132  0855 

35-55 
35-56 

.144  7543 
.144  9704 

36.02 
36-03 

.157  8079 
.158  0269 

36.50 
36.51 

.171  OAOI 

.171  2622 

37.01 

37.02 

60 

2.132  2989 

35-57 

2.145  1866 

36.03 

2.158  2460 

36.52 

2.171  4844 

37.03 

591 


TABLE  VI, 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


r. 

104° 

105° 

106° 

107° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  I". 

logM. 

Diff.  1". 

0' 

1 
2 

2.171  4844 
.171  7066 
.171  9288 

37.03 
37-04 
37-05 

.184  9092 
.185  1346 
.185  3600 

37-56 
37-57 
37-57 

.198  5282 
.198  7568 
.198  9856 

38.11 
38.12 

38.13 

.212  3493 

.212  5814 
.212  8136 

38.68 
38.69 
38.70 

3 
4 

.172  1511 

.172  3735 

37-05 
37.06 

.185  5855 
.185  8110 

37.58 
37-59 

.199  2144 
.199  4432 

38.14 
38.14 

.213  0458 
.213  2781 

38-7I 
38.72 

5 

2.172  5959 

37.07 

2.186  0366 

37.60 

2.199  6721 

38.15 

2.213  5104 

38.73 

6 

.172  8184 

37.08 

.186  2622 

37-6i 

.199  9010 

38.16 

.213  7428 

38.74 

7 

.173  0409 

37.09 

.186  4879 

37.62 

.200  1300 

38.17 

.213  9753 

38-75 

8 

.173  2634 

37.10 

.186  7137 

37.63 

.200  3591 

38.18 

.214  2078 

38-76 

9 

.173  4860 

37.11 

.186  9395 

37.64 

.200  5882 

38.19 

.214  4404 

38.77 

10 

2.173  7087 

37.12 

2.187  1653 

37.65 

2.200  8174 

38.20 

2.214  673° 

38-78 

11 

•173  93*4 

37.12 

.187  3912 

37-66 

.201  0467 

38.21 

.214  9057 

38.79 

12 

.174  1542 

37-13 

.187  6172 

37.67 

.2OI  2760 

38.22 

.215  1385 

38.80 

13 

.174  3770 

37-H 

.187  8432 

37.67 

.201  5053 

38.23 

.215  3713 

38.81 

14 

.174  5999 

37.15 

.188  0693 

37.68 

.201  7347 

38.24 

.215  6042 

38.82 

15 

2.174  8228 

37.16 

2.188  2954 

37.69 

2.2OI  9642 

38.25 

2.215  8371 

38.83 

16 

.175  0458 

37-17 

.188  5216 

37-70 

.202  1937 

38.26 

.216  0701 

38.84 

17 

.175  2688 

37.18 

.188  7478 

37.71 

.202  4233 

38.27 

.216  3032 

38-85 

18 

.175  4919 

37.18 

.188  9741 

37-72 

.202  6529 

38.28 

.216  5363 

38.86 

19 

.175  7150 

37-19 

.189  2005 

37-73 

.202  8826 

38.29 

.216  7694 

38-87 

20 

2.175  9382 

37.20 

2.189  4269 

37-74 

2.203  1123 

38.30 

2.217  0027 

38.88 

21 

.176  1615 

37-21 

.189  6533 

37-75 

.203  3421 

38.31 

.217  2360 

38.89 

22 

.176  3848 

37-22 

.189  8798 

37-76 

.203  5720 

.217  4693 

38.90 

23 

.176  6081 

37-23 

.190  1064 

37-77 

.203  8019 

38*32 

.217  7027 

38.91 

24 

.176  8315 

.190  3330 

37-77 

.204  0319 

38.33 

.217  9362 

38.92 

25 

2.177  0550 

37-25 

2.190  5597 

37.78 

2.204  2619 

38.34 

2.218  1697 

38-93 

26 

.177  2785 

37.25 

.190  7864 

37-79 

.204  4920 

38.35 

.218  4033 

27 

.177  5020 

37.26 

.191  0132 

37.8o 

.204  7222 

38.36 

.218  6369 

38-95 

28 

.177  7256 

37.27 

.191  2401 

37.8i 

.204  9524 

38.37 

.218  8706 

38-96 

29 

•177  9493 

37.28 

.191  4670 

37.82 

.205  1826 

38.38 

.219  1044 

38.97 

30 

2.178  1730 

37-29 

2.191  6939 

37.83 

2.2O5  4.129 

38.39 

2.219  3382 

38-98 

31 

.178  3968 

373° 

.191  9209 

37.84 

.205  6433 

38.40 

.219  5721 

38.99 

32 

.178  6206 

37-31 

.192  1480 

37.85 

•205  8737 

38.41 

.219  8061 

39.00 

33 

.178  8445 

37-32 

.192  3751 

37.86 

.206  1042 

38.42 

.220  0401 

39.01 

34 

.179  0684 

37-33 

.192  6023 

37.87 

.206  3348 

38.43 

.220  2741 

39.02 

35 

2  179  2924 

37-33 

2.192  8295 

37.88 

2.206  5654 

38.44 

2.220  5082 

39-03 

36 

.179  5164 

37-34 

.193  0568 

37-88 

.206  7961 

38-45 

.220  7424 

39-°4 

37 

.179  7405 

37-35 

.193  2841 

37.89 

.207  O268 

38.46 

.220  97^7 

39-05 

38 

.179  9646 

37.36 

.193  5115 

37-9° 

-207  2575 

38.47 

.221  2110 

39.06 

39 

.180  1888 

37-37 

.193  7389 

37-91 

.207  4884 

38.48 

•221  4453 

39.07 

40 

2.l8o  4131 

37.38 

2.193  9664 

37-92 

2.207  7J93 

38.49 

2.221  6797 

39.08 

41 

.l8o  6374 

37-39 

.194  1940 

37-93 

.207  9502 

38.50 

.221  9142 

39.09 

42 

.l8o  8617 

37-4° 

.194  4216 

37-94 

.208  1812 

38.51 

.222  1488 

39.10 

43 

.l8l  o86l 

37-41 

.194  6493 

37-95 

.208  4123 

•222  3834 

39-11 

44 

.l8l  3106 

37.41 

.194  8770 

37-96 

.208  6434 

38-53 

.222  6l8o 

45 

2.181  5351 

37.42 

2.195  1048 

37-97 

2.208  8746 

38.54 

2.222  8528 

39-'3 

46 

.181  7597 

37-43 

.195  3326 

37-98 

.209  1058 

38.54 

.223  0876 

39-H 

47 

.181  9843 

37-44 

.195  5605 

37-99 

.209  3371 

38.55 

.223  3224 

39-'5 

•  48 

.182  1089 

37-45 

.195  7885 

38.00 

.209  5685 

38.56 

.223  5573 

39,16 

49 

-182  4337 

37.46 

.196  0165 

38.00 

•209  7999 

38-57 

.223  7923 

5O 

2.182  6584 

37-47 

2.196  2445 

38.01 

2.210  0314 

38.58 

2.224  0273 

39.18 

51 

.182  8833 

37.48 

.196  4726 

38.02 

.2IO  2629 

38.59 

.224  2624 

39-19 

52 

.183  1082 

37-49 

.196  7008 

38-03 

.210  4945 

38.60 

.224  4975 

39.20 

53 
54 

.183  3331 
.183  5581 

37-49 
37.50 

.196  9290 
.197  1573 

38.04 
38.05 

.210  7261 
.210  9578 

38.61 
38.62 

.224  7327 
.224  9680 

39.21 
39-22 

55 

2.183  7831 

37.51 

2.197  3856 

38.06 

2.21  I  1896 

38.63 

2.225  2033 

39-23 

56 

.184  0082 

37-52 

.197  6140 

38.07 

.211  4214 

38.64 

.225  4387 

39-24 

57 

.184  2334 

37.53 

.197  8425 

38.08 

.211  6533 

38.65 

.225  6741 

3925 

58 

.184  4586 

37-54 

.198  0710 

38.09 

.211  8852 

38.66 

.225  9096 

39.26 

59 

.184  6839 

37-55 

.198  2995 

38.10 

.212  1172 

38.67 

.226  1452 

39-27 

60 

2.184  9092 

37.56 

2.198  5282 

38.11 

2.212  3493 

38.68 

2.226  3808 

39.28 

692 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

108° 

109° 

110° 

111° 

logM. 

Diff.  1". 

log  M. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0 

1 

.226  3808 
.226  6165 

39.28 
39-29 

.240  6314 
.240  8708 

39.90 
39.91 

.255  1099 
•255  353* 

40.54 

40.55 

.269  8255 
.270  0728 

41.21 

2 

.226  8523 

39-3° 

.241  1103 

39-9* 

.255  5965 

40.56 

.270  3202 

41.24 

3 

.227  0881 

39-31 

.241  3498 

3993 

.255  8399 

40.58 

.270  5676 

41.25 

4 

.227  3240 

39.32 

.241  5894 

39-94 

.256  0834 

40.59 

.270  8152 

41.26 

5 

.227  5599 

39-33 

2.241  8291 

39-95 

.256  3270 

40.60 

.271  0628 

41.27 

6 

.227  7959 

39-34 

.242  0688 

39-96 

.256  5706 

40.61 

.271  3104 

41.28 

7 

.228  0320 

39-35 

.242  3086 

39-97 

.256  8143 

40.62 

.271  5582 

41.29 

8 

.228  2681 

39-36 

-242  5485 

39'98 

.257  0580 

40.63 

.271  8060 

41.30 

9 

.228  5043 

39-37 

.242  7884 

39-99 

.257  3019 

40.64 

.272  0538 

41.32 

10 

2.228  7405 

39.38 

2.243  0*84 

40.00 

2.257  5458 

40.65 

2.272  3018 

41-33 

11 

.228  9768 

39-39 

.243  2685 

40.01 

.257  7897 

40.66 

.272  5498 

41-34 

12 

.229  2131 

39-4° 

.243  5086 

40.02 

.258  0337 

40.68 

.272  7979 

41-35 

13 

.229  4496 

39.41 

.243  7488 

40.03 

.258  2778 

40.69 

.273  0460 

41.36 

14 

.229  6861 

39.42 

.243  9890 

40.05 

.258  5220 

40.70 

.273  2942 

41.38 

15 

2.229  92*6 

39-43 

2.144  2293 

40.06 

2.258  7662 

40.71 

2.273  54*5 

41-39 

16 

.230  1592 

39-44 

.244  4697 

40.07 

.259  0105 

40.72 

.273  7909 

41.40 

17 

.230  3959 

39-45 

.244  7101 

40.08 

.259  2548 

40-73 

.274  0393 

41.41 

18 

.230  6326 

39-46 

.244  9506 

40.09 

.259  4992 

4°-74 

.274  2878 

41.42 

19 

.230  8694 

39-47 

.245  1912 

40.10 

•259  7437 

40.75 

.274  5364 

41.43 

20 

2.231  1063 

39-48 

2.245  4318 

40.11 

2.259  9883 

40.76 

2.274  7850 

41.44 

21 

.231  3432 

39-49 

.245  6725 

40.12 

.260  2329 

40.78 

.275  0337 

41.46 

22 

.231  5802 

39-5° 

.245  9132 

40.13 

.260  4776 

40.79 

.275  2825 

41.47 

23 

.231  8172 

39-51 

.246  1541 

40.14 

.260  7223 

40.80 

-275  53*3 

41.48 

24 

.232  0543 

39.52 

.246  3949 

40.15 

.260  9671 

40.81 

.275  7802 

41.49 

25 

2.232  2915 

39-53 

2.246  6359 

40.16 

2.26l  2120 

40.82 

2.276  0292 

41.50 

26 

.232  5287 

39-54 

.246  8769 

40.17 

.26l  4570 

40.83 

.276  2783 

27 

.232  7660 

39-55 

.247  1  1  80 

40.18 

.26l  7020 

40.84 

.276  5274 

41-53 

28 

.233  0033 

39-56 

•*47  3591 

40.19 

.26l  9471 

40.85 

.276  7766 

41.54 

29 

.233  2407 

39-57 

.247  6003 

40.21 

.262  1922 

40.86 

.277  0258 

41-55 

30 

2.233  4782 

39-58 

2.247  8416 

40.22 

2.262  4374 

40.88 

2.277  *752 

41.56 

31 

•233  7*57 

39-59 

.248  0829 

40.23 

.262  6827 

40.89 

.277  5246 

41-57 

32 

•233  9533 

39.60 

.248  3243 

40.24 

.262  9281 

40.90 

.277  7740 

41.58 

33 

.234  1910 

39.61 

.248  5658 

40.25 

.263  1735 

40.91 

.278  0236 

41.60 

34 

.234  4287 

39-63 

.248  8073 

40.26 

.263  4190 

40.92 

.278  2732 

41.61 

35 

2.234  6665 

39.64 

2.249  0489 

40.27 

2.263  6645 

40-93 

2.278  5229 

41.62 

36 

.234  9043 

.249  2906 

40.28  . 

.263  9102 

40.94 

.278  7726 

41.63 

37 

.235  1422 

39.66 

.249  5323 

40.29 

.264  1559 

40.95 

.279  0224 

41.64 

38 

.235  3802 

39-67 

.249  7741 

40.30 

.264  4016 

40.96 

.279  2723 

41.65 

39 

.235  6183 

39-68 

.250  0159 

40.31 

.264  6474 

40.98 

.279  5223 

41.67 

40 

2.235  8563 

39-69 

2.250  2578 

40.32 

2.264  8933 

40.99 

2.279  7723 

41.68 

41 

.236  0945 

39.70 

.250  4998 

40.34 

.265  1393 

41.00 

.280  0224 

41.69 

42 

.236  3327 

39-71 

.250  7419 

40.35 

.265  3853 

41.01 

.280  2726 

41.70 

43 

.236  5710 

39-72 

.250  9840 

40.36 

.265  6314 

41.02 

.280  5228 

41.71 

44 

.236  8093 

39-73 

.251  2262 

4037 

.265  8776 

41.03 

.280  7731 

41.72 

45 

2.237  0478 

39-74 

2.251  4684 

40.38 

2.266  1238 

4I.  OA 

2.281  0235 

41.74 

46 

.237  2862 

39-75 

251  7107 

40.39 

.266  3701 

41.06 

.281  2740 

4*-75 

47 

.237  5247 

39-76 

.251  9531 

40.40 

.266  6165 

41.07 

.281  5245 

41.76 

48 

.237  7633 

39-77 

.252  1955 

40.41 

.266  8629 

41.08 

.281  7751 

41-77 

49 

.238  OO2O 

39.78 

.252  4380 

40.42 

.267  1094 

41.09 

.282  0258 

41.78 

50 
51 
52 

2.238  2407 

•238  4795 
.238  7284 

39-79 
39.80 
39.81 

2.252  6806 
.252  9232 
•253  l659 

4°-43 
40.44 
40.46 

2.267  3560 
.267  6026 
.267  8493 

41.10 
41.11 

41.12 

2.282  2765 
.282  5273 
.282  7782 

41.80 
41.81 
41.82 

53 

•238  9573 

39.82 

•253  4°87 

40.47 

.268  0961 

41.13 

.283  0291 

41.83 

54 

.239  1962 

.253  6515 

40.48 

.268  3430 

4I.I5 

.283  2801 

41.84 

55 

2-239  4353 

39-84 

2.253  8944 

40.49 

2.268  5899 

4I.l6 

2.283  53'* 

41.85 

56 

.239  6744 

39.86 

40.50 

.268  8769 

41.17 

.283  7824 

41.87 

57 

58 
59 

.239  9235 
.240  1528 
.240  3921 

39-87 
39-88 
39-89 

.254  3804 
.254  6235 
.254  8666 

40.51 
40.52 
40.53 

.269  0839 
.269  3310 
.269  5782 

4I.I8 
41.19 

41.20 

.284  0336 
.284  2849 
.284  5363 

41.88 
41.89 
41.90 

60 

2.240  6314 

39.90 

2-255  I099 

40.54 

2.269  8255 

4I.2I 

2.284  7878 

41.91 

38 


593 


TABLE  VI. 

finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V, 

112° 

113° 

114° 

115° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0' 

2.284  7878 

41.91 

2.300  0067 

42.64 

2.315  4927 

43.40 

2.331  2564 

44.18 

1 

.285  0393 

41.93 

.300  2626 

42.65 

•315  7531 

43.41 

.331  5216 

44.20 

2 

.285  2909 

41.94 

.300  5186 

42.67 

.316  0136 

43.42 

.331  7868 

44.21 

3 

.285  5425 

41-95 

.300  7746 

42.68 

.316  2742 

43-44 

.332  0521 

44.22 

4 

•285  7943 

41.96 

.301  0307 

42.69 

.316  5348 

43-45 

•332  3*75 

44.24 

5 

2.286  0461 

41.97 

2.301  2869 

42.70 

2.316  7956 

43.46 

2.332  5830 

44.25 

6 

.286  2979 

41.99 

•301  5431 

42.72 

.317  0564 

43-47 

.332  8485 

44.26 

7 

.286  5499 

42.00 

.301  7995 

42.73 

•3*7  3'73 

43-49 

•333  "4i 

44.28 

8 

.286  8019 

42.01 

.302  0559 

42.74 

.317  5782 

43-5° 

•333  3799 

44.29 

9 

.287  0540 

42.02 

.302  3123 

42.75 

•3i7  8393 

43-51 

•333  6456 

44.31 

10 

2.287  3062 

42.03 

2.302  5689 

42.76 

2.318  1004 

43-53 

2-333  9"5 

44.32 

11 

.287  5584 

42.04 

.302  8255 

42.78 

.318  3616 

43-54 

•334  1775 

44-33 

12 

.287  8107 

42.06 

.303  0822 

42.79 

.318  6229 

43-55 

•334  4435 

44-34 

13 

.288  0631 

42.07 

.303  3390 

42.80 

.318  8842 

43.56 

•334  7°96 

44.36 

14 

.288  3155 

42.08 

•3°3  5958 

42.81 

.319  1456 

43-58 

•334  9758 

44-37 

15 

2.288  5680 

42.09 

2.303  8528 

42.83 

2.319  4072 

43-59 

2.335  2421 

44-39 

16 

.288  8206 

42.10 

.304  1098 

42.84 

.319  6687 

43.60 

•335  5°84 

44.40 

17 
18 

.289  0733 
.289  3260 

42.12 
42.13 

.304  3668 
.304  6240 

42.85 
42.86 

.319  9304 
.320  1921 

43.62 
43-63 

•335  7749 
.336  0414 

44.41 
44-43 

19 

.289  5788 

42.14 

.304  8812 

42.88 

.320  4540 

43-64 

.336  3080 

44.44 

20 

2.289  8317 

42.15 

2-3°5  J385 

42.89 

2.320  7159 

43-66 

2.336  5747 

44-45 

21 

.290  0847 

42.16 

•3°5  3959 

42.90 

.320  9778 

43.67 

.336  8414 

44-47 

22 

.290  3377 

42.18 

•3°5  6533 

42.91 

.321  2399 

43-68 

•337  1-083 

44.48 

23 

.290  5908 

42.19 

.305  9109 

42.93 

.321  5020 

43-69 

•337  3752 

44-49 

24 

.290  8440 

43.20 

.306  1685 

42.94 

.321  7642 

43-70 

.337  6422 

44-51 

25 

2.291  0972 

42.21 

2.306  4261 

42.95 

2.322  0265 

43.72 

2.337  9093 

44.52 

26 

.291  3505 

42.22 

.306  6839 

42.96 

.322  2889 

43-73 

•338  1765 

44-53 

27 

.291  6039 

42.24 

.306  9417 

42.98 

.322  5513 

43-75 

•338  4437 

44-55 

28 

.291  8574 

42.25 

.307  1996 

42.99 

.322  8139 

43-76 

.338  7111 

44-56 

29 

.292  1109 

42.26 

.307  4576 

43.00 

.323  0765 

43-77 

•338  9785 

44-58 

30 

2.292  3645 

42.27 

2.307  7157 

43.02 

2.323  3391 

43-79 

2.339  246o 

44-59 

31 

.292  6182 

42.29 

.307  9738 

43-03 

.323  6019 

43.80 

•339  5*35 

4460 

32 

.292  8719 

4230 

.308  2320 

43  -°4 

.323  8647 

43.81 

•339  7812 

44.62 

33 

.293  1258 

4*.  3  * 

.308  4903 

43.05 

.324  1277 

43-83 

•34°  °49° 

44.63 

34 

•293  3797 

42.32 

.308  7486 

43-°7 

.324  3907 

43.84 

.340  3168 

44.64 

35 

2.293  6336 

42.33 

2.309  0071 

43.08 

2.324  6537 

43-85 

2.340  5847 

44.66 

36 

.293  8877 

4*-  3  5 

.309  2656 

43.09 

.324  9169 

43-87 

.340  8527 

44h7 

37 

.294  1418 

42.36 

.309  5242 

43.10 

.325  1801 

43-88 

.341  1207 

44.69 

38 

.294  3960 

42.37 

.309  7828 

43.12 

•3*5  4434 

43-89 

.341  3889 

44.70   ( 

39 

.294  6503 

42.38 

.310  0416 

43-  '  3 

.325  7068 

43-91 

.341  6571 

44-71   j 

40 

2.294  9046 

42.40 

2.310  3004 

43-H 

2.325  9703 

43-9* 

2-341  9255 

44-73 

41 

.295  1590 

42.41 

•310  5593 

43-15 

.326  2339 

43-93 

.342  1939 

44-74 

42 

•295  4135 

42.42 

.310  8182 

43-J7 

.326  4975 

43-94 

.342  4623 

44-75 

43 

.295  6680 

42.43 

.311  0773 

43.18 

.326  7612 

43.96 

.342  7309 

44-77 

44 

.295  9227 

42.44 

•3"  3364 

43-J9 

.327  0250 

43-97 

•342  9995 

44.78 

45 

2.296  1774 

42.46 

2.311  5956 

43.21 

2.327  2889 

43-98 

2-343  2683 

44.80 

46 

.296  4321 

42.47 

.311  8549 

43.22 

.327  5528 

44.00 

•343  537i 

44.81 

47 

.296  6870 

42.48 

.312  1142 

43.23 

.327  8168 

44.01 

.343  8060 

44.82 

48 

.296  9419 

42.49 

.312  3736 

43-24 

.328  0809 

44.02 

•344  0750 

44.84 

49 

.297  1969 

42.51 

.312  6331 

43.26 

.328  3451 

44.04 

•344  344° 

44.85 

50 

2.297  4520 

42.52 

2.312  8927 

43.27 

2.328  6094 

44.05 

2-344  6132 

*-£ 

51 

.297  7071 

42.  5  3 

.313  1524 

43.28 

.328  8737 

44.06 

.344  8824 

44.88 

52 

.297  9623 

42.54 

.313  4121 

43.29 

.329  1382 

44.08 

•345  MI7 

44.89 

53 

.298  2176 

42-55 

.313  6719 

43-31 

.329  4027 

44.09 

•345  4211 

44.91 

54 

.298  4730 

42-57 

•3i3  93l8 

43-32 

.329  6672 

44.10 

.345  6906 

44.92 

55 
56 

2.298  7284 
.298  9839 

42.58 
42.59 

2.314  1917 
.314  4518 

43-33 
43-35 

2.329  9319 
.330  1967 

44.12 

44-  '  3 

2.345  9601 
.346  2298 

44-93 
44-95 

57 

.299  2395 

42.60 

.314  7119 

43-36 

•33°  4615 

44.14 

.346  4995 

44.96 

58 

.299  4952 

42.61 

.314  9721 

43-37 

.330  7264 

44.16 

•346  7693 

44-97 

59 

.299  7509 

42.63 

.315  2323 

43.38 

•33°  99J4 

44.17 

•347  0392 

44-99 

60 

2.300  0067 

42.64 

2.315  4927 

43.40 

2.331  2564 

44.18 

2-347  3092 

45.00 

694 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

116° 

117° 

118° 

119° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0' 

2.347  3092 

45.00 

2.363  6626 

45.86 

2.380  3290 

46.74 

1.397  3210 

47.66 

1 

•347  5792 

45.02 

•363  9378 

45.87 

.380  6095 

46.76 

.397  6070 

47.68 

2 

•347  8494 

45-°3 

.364  2131 

45.88 

.380  8901 

46.77 

•397  8931 

47.70 

1   3 

.348  1196 

45.04 

.364  4885 

45.90 

.381  1708 

46.79 

.398  1794 

47-71 

4 

•348  3899 

45.06 

.364  7639 

45.91 

•381  45*5 

46.80 

•398  4657 

47-73 

5 

2.348  6603 

45-07 

2-365  °394 

45-93 

1.381  7324 

46.82 

2.398  7521 

47-74 

6 

.348  9308 

45-09 

•365  3^5° 

45-94 

.382  0133 

46.83 

•399  °386 

47.76 

7 

.349  2014 

45.10 

-365  5907 

45.96 

.382  2944 

46.85 

-399  3252 

47-77 

8 

•349  4720 

45.11 

.365  8665 

45-97 

.382  5755 

46.86 

•399  6119 

47-79 

9 

•349  7428 

45-13 

.366  1423 

45-99 

.382  8567 

46.88 

•399  8987 

47.81 

10 

2350  0136 

45.14 

2.366  4183 

46.00 

2.383  1380 

46.89 

2.400  1856 

47-82 

11 

•35°  2845 

45.16 

.366  6944 

46.01 

.383  4i94 

46.91 

.400  4725 

47.84 

12 

•35°  5554 

45-17 

.366  9705 

46.03 

.383  7009 

46.92 

.400  7596 

47.85 

13 

.350  8265 

45  i8 

.367  2467 

46.04 

.383  9825 

46.94 

.401  0468 

47.87 

14 

.351  0977 

45.20 

.367  5230 

46.06 

.384  2642 

46.95 

.401  3340 

47.89 

15 

2351  3689 

45.21 

2,367  7994 

46.07 

2.384  5460 

46.97 

2.401  6214 

47.90 

16 

.351  6402 

45-^3 

.368  0759 

46.09 

.384  8278 

46.98 

.401  9088 

47.92 

17 

.351  9116 

45.24 

.368  3525 

46.10 

.385  1098 

46.99 

.402  1964 

47-93 

18 

.352  1831 

45-25 

.368  6291 

46.12 

.385  3918 

47.01 

.402  4840 

47-95 

19 

•35*  4547 

45.27 

.368  9059 

46.13 

•385  6739 

47.03 

.402  7718 

47-97 

20 

2.352  7263 

45-28 

2.369  1827 

46.15 

2.385  9562 

47.05 

2.403  0596 

47.98 

21 

.352  9981 

45-3° 

.369  4596 

46.16 

.386  2385 

47.06 

•4°3  3475 

48.00 

22 

•353  2699 

45.31 

.369  7367 

46.18 

.386  5209 

47.08 

•4°3  6356 

48.01 

23 

•353  54l8 

45-33 

.370  0138 

46.19 

.386  8034 

47.09 

•403  9237 

48.03 

24 

•353  8138 

45-34 

.370  2909 

46.21 

.387  0860 

47.11 

.404  2119 

48.04 

25 

2.354  °859 

45-35 

2.370  5682 

46.22 

2.38-7  3687 

47.12 

2.404  5002 

48.06 

2G 

•354  358i 

45-37 

.370  8456 

46.24 

•387  6514 

47.14 

.404  7886 

48.08 

27 

•354  63°3 

45-38 

.371  1230 

46.25 

-387  9343 

47-15 

•4°5  O77i 

48.09 

28 

.354  9027 

45.40 

.371  4006 

46.26 

.388  2173 

47-*7 

•4°5  3657 

48.11 

29 

•355  '751 

45-41 

.371  6782 

46.28 

.388  5003 

47.18 

.405  6544 

48.12 

30 

2-355  4476 

45-42 

2-37'  9559 

46.29 

2-388  7835 

47.20 

2.405  9432 

48.H 

31 

.355  7202 

45-44 

.372  2337 

46.31 

.389  0667 

47.21 

.406  2321 

48.16 

32 

•355  9928 

45-45 

•372  5Ij6 

46.32 

•389  35°° 

47-23 

.406  5211 

48.17 

33 

.356  2656 

45-47 

-372  7896 

46-34 

•389  6335 

47.24 

.406  8102 

48.19 

34 

•356  5385 

45-48 

.373  0677 

46.35 

•389  9*70 

47.26 

•4°7  0993 

48.20 

35 

2.356  8114 

45-5° 

2-373  3459 

46-37 

2.390  2006 

47.28 

2.407  3886 

48.22 

36 

•357  °844 

45-51 

•373  624* 

46.38 

.390  4843 

47.29 

.407  6780 

48.24 

37 

•357  3575 

45-52 

•373  9024 

46.40 

.390  7681 

47-31 

.407  9674 

48.25 

38 

•357  6307 

45-54 

.374  1809 

46.41 

.391  0519 

47-32 

.408  2570 

4*'*l 

39 

•357  9040 

45-55 

•374  4594 

46.43 

•391  3359 

47-34 

.408  5467 

48.28 

40 

2.358  1773 

45-57 

2-374  738° 

46.44 

2.391  6200 

47-35 

2.408  8364 

48.30 

41 

.358  4508 

45.58 

•375  Ol67 

46.46 

".392  9042 

47-37 

.409  1263 

48.32 

42 

.358  7243 

45.60 

•375  2955 

46.47 

.392  1884 

47-38 

.409  4162 

48.33 

43 

.358  9979 

45.61 

•375  5744 

46.49 

-392  4728 

47.40 

.409  7063 

48.35 

44 

•359  2716 

45.62 

•375  8533 

46.50 

.392  7572 

47.41 

.409  9964 

48.37 

45 

2-359  5454 

45.64 

2.376  1324 

46.51 

*-393  °4J7 

47-43 

2.410  2866 

48.38 

46 

•359  8l93 

45.65 

.376  4115 

46.53 

•393  3264 

47-45 

.410  5770 

48.40 

47 

.360  0933 

45.67 

.376  6908 

46.55 

•393  6l11 

47-46 

.410  8674 

48.41   , 

48 

.360  3673 

45.68 

.376  9701 

46.56 

•393  8959 

47.48 

.411  1579 

48.43 

49 

.360  6415 

45.70 

•377  2495 

46..58 

.394  1808 

47-49 

.411  4486 

48-45 

50 

2.360  9157 

45-71 

2-377  529° 

46.59 

2-J94  4658 

47.51 

2.411  7393 

48.46 

51 

.361  1900 

45-72 

.377  8086 

46.60 

•394  75°9 

47.52 

.412  0301 

48.48 

52 

.361  4644 

45-74 

.378  0883 

46.62 

-395  °361 

47-54 

.412  3210 

48.49 

53 
54 

.361  7389 
.362  0134 

45-75 
45-77 

.378  3681 
•378  6479 

46.64 
46.65 

•395  3214 
•395  6067 

47-55 
47-57 

.412  0120 
.412  9031 

48.51 
48.53 

55 

2.362  2881 

45-78 

2-378  9279 

46.67 

2-395  8922 

47-59 

2.413  1944 

4s'5* 

56 

.362  5628 

45.80 

•379  2079 

46.68 

.396  1778 

47.60 

.413  4857 

48-56 

57 

.362  8376 

45.81 

•379  4881 

46.70 

.396  4634 

47.62 

.413  7771 

48.58 

58 
59 

.363  1126 
.363  3876 

45.82 
45.84 

•379  7683 
.380  0486 

46.71 
46-73 

•396  7492 
•397  0350 

47-63 
47.65 

.414  0686 
.414  3602 

48-59 
48.61 

60 

2.363  6626 

45-86 

2.380  3290 

46.74 

2.397  3210 

47.66 

2.414  6519 

48.62 

695 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

120° 

121° 

122° 

123° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0 

2.414  6519 

48.62 

M-3*  3356 

49.62 

2.450  3868 

50.67 

2.468  8205 

5'-75 

1 

.414  9437 

48.64 

•43*  6334 

49.64 

.450  6908 

50.68 

.469  1311 

51-77 

2 

.415  2356 

48.66 

.432  9313 

49.66 

.450  9950 

50.70 

.469  4418 

51-79 

3 

.415  5276 

48.67 

•433  2293 

49.68 

•45  *  2992 

50.72 

.469  7526 

5l.8l 

4 

.415  8197 

48.69 

•433  5*74 

49.69 

.451  6036 

5°-74 

.470  0634 

51.82 

5 

2.416  1119 

48.71 

2-433  8257 

49.71 

2.451  9081 

5°-75 

2.470  3744 

51.84 

6 

.416  4042 

48.72 

.434  1240 

49-73 

.452  2127 

50.77 

.470  6856 

51.86 

7 
8 

.416  6965 
.416  9890 

48.74 
48.76 

•434  42*4 
.434  7209 

49-74 
49.76 

•45*  5174 
.452  8222 

50.79 
50.81 

•47°  99^8 
.471  3081 

51.88 
51.90 

9 

.417  2816 

48.77 

•435  °i95 

49.78 

•453  1271 

50.83 

.471  6196 

51.92 

10 

2.417  5743 

48.79 

2.435  3182 

49.80 

2.453  432i 

50.84 

2.471  9311 

51.94 

11 

.417  8671 

48.81 

•435  6171 

49.81 

•453  7372 

50.86 

.472  2428 

5*-95 

12 

.418  1600 

48.82 

•435  916° 

49.83 

•454  04M 

50.88 

.472  5546 

51-97 

13 

.418  4529 

48.84 

.436  2150 

49.85 

•454  3477 

50.90 

.472  8665 

51.99 

14 

.418  7460 

48.85 

.436  5141 

49.86 

•454  6532 

50.92 

•473  J785 

52.01 

15 

2.419  0392 

48-87 

1.436  8134 

49.88 

2.454  9587 

5°-93 

2.473  49°6 

52-03 

16 

•4i9  3325 

48.89 

*437  "27 

49.90 

.455  2644 

50.95 

.473  8028 

52-05 

17 

.419  6258 

48.90 

.437  4122 

49.92 

•455  57°i 

50.97 

•474  "5* 

52-07 

18 

.419  9193 

48.92 

•437  7ii7 

49-93 

.455  8760 

50.99 

.474  4276 

52.09 

19 

.420  2129 

48.94 

.438  0114 

4995 

.456  1820 

51.00 

•474  74°2 

52.10 

20 

4.420  5066 

48.95 

2.438  3111 

49-97 

2.456  4881 

51.02 

2.475  0529 

52.12 

21 

.420  8003 

48.97 

.438  6110 

49.98 

.456  7943 

51.04 

•475  3657 

52.14 

22 

.421  0942 

48.99 

.438  9109 

50.00 

.457  1006 

51.06 

.475  6786 

52.16 

23 

.421  3882 

49.00 

.439  21  10 

50.02 

.457  4070 

51.08 

•475  99l6 

52.18 

24 

.421  6822 

49.02 

•439  5"2 

50.04 

•457  7135 

51.09 

.476  3047 

52.20 

25 

2.421  9764 

49-°3 

2.439  8114 

50.05 

2.458  0201 

51.11 

2.476  6180 

52.22 

26 

.422  2707 

49.05 

.440  1118 

50.07 

.458  3268 

51.13 

.476  9313 

52.23 

27 

.422  5650 

49.07 

.440  4123 

50.09 

•458  6337 

51.15 

.477  2448 

52.25 

28 

.422  8595 

49.09 

.440  7129 

50.1  1 

.458  9406 

51.17 

•477  5584 

52.27 

29 

.423  1541 

49.10 

.441  0136 

50.12 

•459  2477 

51.18 

.477  8721 

52.29 

30 

2.423  4488 

49.12 

2.441  3143 

50.14 

2.459  5548 

51.20 

2.478  1859 

52-31 

31 

.423  7435 

49.14 

.441  6152 

50.16 

.459  8621 

51.22 

.478  4998 

52-33 

32 

.424  0384 

49.15 

.441  9162 

50.18 

.460  1695 

51.24 

.478  8138 

52-35 

33 

.424  3334 

49.17 

.442  2173 

50.19 

.460  4770 

51.26 

.479  1280 

52  37 

34 

.424  6284 

49.19 

.442  5185 

50.21 

.460  7846 

51.28 

•479  4422 

52.39 

35 

2.424  9236 

49.20 

2.442  8199 

50.23 

2.461  0923 

51.29 

2.479  7566 

52.40 

36 

.425  2189 

49.22 

.443  1213 

50.24 

.461  4001 

5M1 

.480  0711 

52.42 

37 

.425  5142 

49.24 

.443  4228 

50.26 

.461  7080 

5'-33 

.480  3857 

52.44 

38 

.425  8097 

49.25 

.443  7244 

50.28 

.462  0161 

51-35 

.480  7004 

52.46 

39 

.426  1053 

49.27 

.444  0261 

50.30 

.462  3242 

5'-37 

.481  0152 

52-48 

40 

2.426  4010 

49.29 

2.444  3280 

50.31 

2.462  6325 

51.38 

2.481  3301 

52.50 

41 

.426  6967 

49-3° 

.444  6299 

5°-33 

.462  9408 

51.40 

.481  6452 

52.52 

42 

.426  9926 

49.32 

.444  9320 

5°-35 

.463  2493 

51.42 

.481  9604 

52-54 

43 

.427  1886 

49-34 

•445  234i 

5037 

•463  5579 

5M4 

.482  2756 

52.56 

44 

.427  5847 

49-35 

•445  5364 

50.38 

.463  8666 

51.46 

.482  5910 

52-58 

45 

2.427  8808 

49-37 

2.445  8387 

50.40 

2.464  1754 

51.48 

2.482  9065 

52.59 

46 

.428  1771 

49-39 

.446  1412 

50.42 

.464  4843 

51.49 

.483  2222 

52.61 

47 

.428  4735 

49.40 

.446  4437 

50.44 

•464  7933 

5*-5« 

•483  5379 

52.63 

!  48 

.428  7700 

49.42 

.446  7464 

50.45 

.465  1024 

5'-53 

•483  8537 

52.65 

49 

.429  0665 

49-44 

.447  0492 

50.47 

.465  4116 

51-55 

.484  1697 

52.67 

50 

2.429  3632 

49.46 

4.447  3521 

50.49 

2.465  7210 

51-57 

2.484  4858 

52.69 

51 

.429  6600 

49-47 

•447  655i 

50.51 

.466  0305 

51-59 

.484  8020 

52.71 

52 

.429  9569 

49-49 

,447  9582 

5°-53 

.466  3400 

51.60 

.485  1183 

52.73 

53 

.430  2539 

49-51 

.448  2614 

50.54 

.466  6497 

51.62 

•485  4347 

52-75 

54 

.430  5510 

49.52 

.448  5647 

50.56 

.466  9595 

51.64 

.485  7513 

52-77 

55 

2.430  8482 

49-54 

2.448  8681 

50.58 

2.467  2694 

51.66 

2.486  0679 

52.78 

56 

.431  1455 

49.56 

•449  i7i6 

50.60 

.467  5794 

51.68 

.486  3847 

52.80 

57 

.431  4418 

49-57 

.449  4753 

50.61 

.467  8895 

51.70 

.486  7016 

52.82 

58 

.431  7403 

49-59 

•449  779° 

50.63 

.468  1997 

51.71 

.487  0186 

52.84 

59 

.432  0379 

49.61 

.450  0828 

50.65 

.468  5101 

5'-73 

•487  3357 

52.86 

60 

2.432  3356 

49.62 

2.450  3868 

50.67 

2.468  8205 

51-75 

2.487  6529  1  52.88 

596 


TABLE  VI, 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit 


V. 

124° 

125° 

126° 

127° 

log  M. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0 

2.487  6529 

52.88 

2.506  9006 

54.06 

2.526  5813 

55-29 

2.546  7135 

56-57 

1 

.487  9702 

52.90 

.507  2251 

54.08 

•526  9131 

55-31 

•547  0530 

56-59 

2 

.488  2877 

52.92 

.507  5496 

54.10 

.527  2450 

55-33 

•547  3926 

56.61 

3 

.488  6053 

52.94 

.507  8742 

54.12 

•527  5771 

55-35 

547  7323 

56-63 

4 

.488  9230 

52.96 

.508  1990 

54.14 

.527  9092 

55-37 

548  0722 

56-65 

5 

2.489  2408 

52.98 

2.508  5239 

54.16 

2.528  2415 

55-39 

2.548  4122 

56.68 

6 

.489  5587 

53.00    .508  8489   54.18 

•528  5739 

55-41 

•548  7523 

56.70 

7 

.489  8767 

53.02 

.509  1741 

54.20 

.528  9065 

55-43 

.549  0926 

56.72 

8 

.490  1949 

53.03 

•5°9  4993 

54-22 

.529  2391 

55-45 

•549  433° 

56.74 

9 

.490  5132 

53-05 

.509  8247 

54.24 

•529  57»9 

55.48 

•549  7735 

56.76 

10 

2.490  8315 

53.07 

2.510  1502 

54-26 

2.529  9048 

55-50 

2.550  1141 

56.79 

11 

491  1500 

53-°9 

.510  4758 

54.28 

•53°  2379 

55-52 

•55°  4549 

56.81 

12 

.491  4686 

53.11 

.510  8016 

54-30 

-53°  57io 

55-54 

•55°  7958 

56.83 

13 

.491  7874 

53-13 

.511  1274 

54-32 

•53°  9043 

55-56 

.551  1369 

56.85 

14 

.492  1063 

53-15 

•511  4534 

54-34 

.531  2378 

55-58 

•551  478i 

56.87 

15 

2.492  4252 

53-17 

2.511  7795 

54.36 

2-531  5713 

55.60 

2.551  8194 

56.90 

1C 

.492  7443 

53-19 

.512  1057 

54-38 

.531  9050 

55.62 

.552  1608 

56.92 

17 

•493  °635 

53-21 

.512  4321 

54.40 

.532  2388 

55.64 

.552  5024 

56.94 

18 

.493  3828 

53-^3 

.512  7586 

54-42 

•532  5727 

55-67 

•552  8441 

56.96 

19 

•493  7023 

53-^5 

.513  0852 

54-44 

.532  9068 

55-69 

•553  1859 

56-98 

20 

2.494  0218 

53.27 

2.513  4119 

54.46 

2-533  2410 

55-71 

2-553  5279 

57.01 

21 

•494  3415 

53-29 

•5i3  7387 

54.48 

•533  5753 

55-73 

•553  8700 

57.03 

22 

•494  66l3 

53-31 

.514  0657 

54-50 

•533  9°97 

55-75 

.554  2122 

57.05 

23 

.494  9812 

53-33 

.514  3927 

54-52 

•534  2443 

55-77 

•554  5546 

57.07 

24 

•495  3°i2 

53-35 

.514  7199 

54-54 

•534  579° 

55-79 

•554  8971 

57.10 

25 

2.495  62i3 

53-37 

2.515  0473 

54.56 

2.534  9138 

55.81 

2-555  2398 

57-12 

2G 

.495  9416 

53-39 

•5*5  3747 

54-58 

•535  2487 

55.84 

•555  5825 

57.H 

27 

28 
29 

.496  2619 
.496  5824 
.496  9030 

53-4i 
53.42 

53-44 

.515  7023 
.516  0300 
.516  3578 

54.60 
54-63 
54-65 

•535  5838 
•535  9*9° 
•536  2543 

55.86 
55-88 
55.90 

•555  9254 
.556  2685 
.556  6116 

57.16 
57.18 
57-21 

30 

2.497  2238 

53-46 

2.516  6857 

54.67 

2.536  5898 

55-92 

2-556  9549 

57-23 

31 

.497  5446 

53.48 

.517  0138 

54.69 

.536  9254 

55-94 

•557  2984 

57-25 

32 

.497  8656 

53-5° 

.517  3420 

54-7i 

.537  2611 

55-96 

•557  6420 

57*27 

33 

.498  1867 

53-52 

.517  6703 

54-73 

•537  5970 

55-98 

•557  9857 

57-29 

34 

.498  5079 

53-54 

.517  9987 

54-75 

•537  9329 

56.01 

•558  3295 

57.32 

35 

2.498  8292 

53-56 

2.518  3273. 

54-77 

2.538  2690 

56.03 

2.558  6735 

57-34 

36 

.499  1506 

53-58 

.518  6559 

54-79 

.538  6052 

56-05 

•559  OI76 

57.36 

37 

.499  4721 

53.60 

.518  9847 

54-8i 

.538  9416 

56.07 

•559  36lg 

57.38 

38 

•499  7938 

53.62 

•5*9  3*37 

54.83 

•539  2781 

56.09 

•559  7062 

57.41 

39 

.500  1156 

53-64 

.519  6427 

54.85 

•539  6i47 

56.11 

.560  0507 

57-43 

40 

2.500  4375 

53.66 

2.519  9719 

54.87 

2-539  95H 

56.13 

2.560  3953 

57-45 

41 

.500  7595 

53.68 

.520  3012 

54.89 

.540  2883 

55-I5> 

.560  7401 

57-47 

42 

.501  0817 

53-70 

.520  6306 

54-91 

.540  6253 

56.18 

.561  0850 

57-50 

43 

.501  4039 

53-72 

.520  9601 

54-93 

.540  9625 

56.20 

.561  4301 

57-52 

44 

.501  7263 

53-74 

.521  2898 

54-95 

.541  2997 

56.22 

.561  7753 

57-54 

45 
46 

47 

2.502  0488 
.502  3714 
.502  6942 

53-76 
53-78 
53.80 

2.521  6196 
.521  9495 
.522  2795 

54-97 
54-99 
55.02 

2.541  6371 
.541  9746 
•542  3*23 

56.24 
56.26 
56.29 

2.562  1206 
.562  4660 
.562  8116 

57-56 

57-59 
57-6i 

48 

.503  0170 

53.82 

.522  6097 

55-04 

•542  6500 

56.31 

•563  '574 

57-63 

49 

.503  3400 

53-84 

.522  9400 

55.06 

.542  9880 

56.33 

•563  5032 

57.65 

50 

2.503  6631 

53.86 

2.523  2704 

55.08 

2-543  3260 

56.35 

2.563  8492 

57.68 

51 

.503  9863 

53.88 

.523  6009 

55.10 

•543  6641 

56-37 

•564  !953 

57-70 

52 
53 
54 

.504  3096 

•504  6331 
.504  9567 

53-9° 
53.92 

53-94 

.523  9316 
.524  2624 
•524  5933 

55-12 

55-14 
55.16 

.544  0024 
•544  34°9 
•544  6794 

56.39 
56.42 
56.44 

•564  54i6 
.564  8880 

•565  2345 

57-72 
57-74 
57-77 

55 
56 

2.505  2804 
.505  6042 

53-96 
53.98 

2.524  9243 
•525  2555 

55.18 

55.20 

2.545  0181 
•545  3569 

56.46 
56.48 

2.565  5812 
.565  9280 

57-79 
57-81 

57 

58 
59 

.505  9282 
.506  2522 
.506  5763 

54.00 
54.02 
54.04 

-525  5867 
.525  9181 
.526  2497 

55-22 

55.24 
55-26 

•545  6959 
.546  0350 
.546  3742 

56.50 
56-52 
56-55 

.566  2750 
.566  6221 
.566  9693 

57.84 
57-86 
57-88 

6O 

2.506  9006 

54.06 

2.526  5813 

55-29 

2-546  7135 

56-57 

2.567  3166 

57.90 

597 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

128° 

129° 

130° 

131° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0' 

2.567  3166 

57.90 

2.588  4112 

59-3° 

2.610  0188 

60.75 

2.632  1622 

62.28 

1 

.567  6641 

57-93 

.588  7670 

59-3* 

.610  3834 

60.78 

.632  5360 

62.30 

2 

.568  0117 

57-95 

.589  1230 

59-35 

.610  7481 

60.80 

.632  9099 

62.33 

3 

.568  3595 

57-97 

.589  4792 

59-37 

.611  1130 

60.83 

.633  2839 

62.35 

4 

.568  7074 

57-99 

.589  8355 

59-39 

.611  4781 

60.85 

.633  6581 

62.38 

5 
6 

2.569  0554 
.569  4036 

58.02 

58.04 

2.590  1919 
.590  5485 

59.42 
59-44 

2.611  8433 
.612  2086 

60.88 
60.90 

2.634  0325 
.634  4070 

62.41 
62.43 

7 

.569  7519 

58.06 

.590  9052 

59-47 

.612  5741 

60.93 

•634  78l7 

62.46 

8 

.570  1004 

58.09 

.591  2620 

59-49 

.612  9397 

60.95 

.635  1565 

62.48 

9 

.570  4490 

58.11 

.591  6190 

59-51 

.613  3055 

60.98 

•635  5315 

62.51 

10 

2.570  7977 

58.13 

2.591  9762 

59-54 

2.613  6715 

6  1.  oo 

2.635  9°66 

62.54 

11 
12 

.571  1465 
•571  4955 

58-15 
58.18 

•59*  3335 
.592  6909 

59.56 
59-58 

.614  0376 
.614  4038 

61.03 
61.05 

.636  2819 
.636  6573 

62.56 
62.59 

13 

.571  8447 

58.20 

•593  °485 

59.61 

.614  7702 

61.08 

.637  0329 

62.61 

14 

.572  1939 

58,22 

.593  4062 

.615  1368 

61.10 

.637  4087 

62.64 

15 

*-57*  5434 

58.25 

2.593  7641 

59.66 

2.615  5035 

61.13 

2.637  7846 

62.67 

16 

.572  8929 

58.27 

.594  1221 

59.68 

.615  8703 

61.15 

.638  1607 

62.69 

17 

.573  2426 

58.29 

•594  4803 

59.70 

.616  2373 

61.18 

•638  5369 

62.72 

18 

•573  59*4 

58.32 

•594  8386 

59-73 

.616  6045 

61.20 

•638  9133 

62.75 

19 

•573  94*4 

58.34 

•595  '97° 

59-75 

.616  9718 

61.23 

.639  2899 

62.77 

20 

2.574  2925 

58.36 

*-595  5556 

59-78 

2.617  3392 

61.25 

2.639  6666 

62.80 

21 

•574  6427 

58-38 

•595  9H3 

59.80 

.617  7068 

61.28 

.640  0435 

62.82 

22 
23 

•574  9931 
•575  3436 

58.41 
58.43 

.596  *73* 
.596  6322 

59.82 
59.85 

.618  0746 
.618  4425 

61.30 

61.33  . 

.640  4205 
.640  7977 

62.85 
62.88 

24 

•575  6943 

58.45 

.596  9914 

59.87 

.618  8105 

61.36 

.641  1750 

62.90 

25 

2.576  0451 

58-48 

2.597  3507 

59-9° 

2.619  *787 

61.38 

2.641  5525 

62.93 

26 

.576  3960 

58.50 

.597  7102 

59.92 

.619  5471 

61.41 

.641  9302 

62.96 

27 

.576  7471 

58.52 

.598  0698 

59-95 

.619  9156 

61.44 

.642  3080 

62.98 

28 

•577  0983 

58.55 

.598  4295 

59-97 

.620  2843 

61.46 

.642  6860 

63.01 

29 

•577  4496  1  58.57 

.598  7894 

59-99 

.620  6531 

61.48 

.643  0641 

63.04 

30 

2.577  8011 

58.59 

2.599  J494 

60.02 

2.621  0220 

61.51 

2.643  44*4 

63.06 

31 

.578  1528 

58.62 

•599  5°96 

60.04 

.621  3911 

61.53 

.643  8209 

63.09 

32 

•578  5°45 

58.64 

.599  8699 

60.07 

.621  7604 

61.56 

.644  1995 

63.12 

33 

•578  8564 

58.66 

.600  2304 

60.09 

.622  1298 

61.58 

.644  5783 

63.14 

34 

.579  2085 

58.69 

.600  5910 

60.12 

.622  4994 

61.61 

.644  9572 

63.17 

35 

2.579  5607 

58.71 

2.600  9518 

60.14 

2.622  8691 

61.63 

2-645  3363 

63.19 

36 

•579  9I3° 

58.73 

.601  3127 

60.  1  6 

.623  2390 

61.66 

.645  7155 

63.22 

37 

.580  2655 

58.76 

.601  6738 

60.19 

.623  6091 

61.68 

.646  0949 

63.25 

38 

.580  6181 

58-78 

.602  0350 

60.21 

•6*3  9793 

61.71 

.646  4745 

63.27 

39 

.580  9708 

58.80 

.602  3963 

60.24 

.624  3496 

61.74 

.646  8542 

63.30 

40 

2.581  3237 

58.83 

2.602  7578 

60.26 

2.624  7*01 

61.76 

2.647  2341 

63-33 

41 

.581  6768 

58.85 

.603  1195 

60.29 

.625  0907 

61.79 

.647  6142 

63.35 

42 
43 

.582  0299 
.582  3832 

58-87 
58.90 

.603  4813 
.603  8432 

60.31 
60.34 

.625  4615 
.625  8325 

61.81 
61.84 

.647  9944 
.648  3748 

63.38 
63.41 

44 

.582  8267 

58.92 

.604  2053 

60.36 

.626  2036 

61.86 

•648  7553 

63.44 

45 

2.583  0903 

58.94 

2.604  5675 

60.38 

2.626  5748 

61.89 

2.649  J36o 

63.46 

46 

.583  4440 

58.97 

.604  9299 

60.41 

.626  9462 

61.91 

.649  5168 

63.49 

47 

•583  7979 

58.99 

.605  2924 

60.43 

.627  3178 

61.94 

.649  8978 

63.52 

48 

.584  1519 

59.01 

.605  6551 

60.46 

.627  6895 

61.97 

.650  2790 

63.54 

49 

.584  5061 

59.04 

.606  0179 

60.48 

.628  0614 

61.99 

.650  6603 

50 

2.584  8604 

59.06 

2.606  3809 

60.51 

2.628  4334 

62.02 

2.651  0418 

63.60 

51 

.585  2148 

59.09 

.606  7440 

60.53 

.628  8056 

62.04 

.651  4235 

63.62 

1  52 

•585  5694 

59.11 

.607  1073 

60.56 

.629  1780 

62.07 

.651  8053 

63-65 

53 

.585  9241 

.607  4707 

60.58 

•6*9  55°5 

62.09 

.652  1873 

63.68 

54 

.586  2790 

59.16 

•607  8343 

60.61 

.629  9231 

62.12 

.652  5695 

63.70 

55 

2.586  6340 

59.18 

2.608  1980 

60.63 

2.630  2959 

62.15 

2.652  9518 

61-73 

56 

.586  9891 

59.20 

.608  5618 

60.66 

.630  6689 

62.17 

•653  334* 

6j.70 

57 

•587  3444 

59-*3 

.608  9258 

60.68 

.631  0420 

62.20 

.653  7168 

63.79 

58 

.587  6999 

59-*5 

.609  2901 

60.70 

.631  4152 

62.22 

.654  0996 

63.81 

59 

-588  0555 

59-*7 

.609  6544 

60.73 

.631  7887 

62.25 

.654  4826 

63.84 

60 

2.588  4112 

59-3° 

2.610  0188 

60.75 

2.632  i6i2 

62.28 

2.654  8657 

63.87 

TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

132° 

133° 

134° 

135° 

log  M. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

o 

.654  8657 

63.87 

2.678  1547 

65-53 

2.702  0562 

67.27 

2.726  5990 

69.09 

1 

.655  2490 

63.89 

.678  5480 

65.56 

.702  4600 

67.30 

.727  0137 

69.12 

2 
3 

.655  6324 
.650  oioo 

63.92 
63-95 

.678  9414 
•679  335° 

65.59 
65.61 

.702  8638 
.703  2679 

67-33 
67.36 

.727  4285 
•7^7  84^5 

69.15 
69.19 

4 

.656  3998 

63.97 

.679  7288 

65.64 

.703  6721 

67-39 

.728  2587 

69.22 

5 

1.656  7837 

64.00 

2.680  1227 

65.67 

2.704  0766 

67.42 

2.728  6741 

69.25 

6 

.657  1678 

64.03 

.680  5168 

65.70 

.704  4812 

67.45 

.729  0897 

69.28 

7 
8 

.657  5521 
.657  9365 

64.06 
64.08 

.680  9111 
.681  3056 

65.73 
65.76 

.704  8860 
.705  2909 

67.48 
67.51 

.729  5055 
.729  9215 

69.31 
69-34 

9 

.658  3211 

64.11 

.681  7002 

65-79 

.705  6961 

67.54 

.730  3376 

69.37 

10 
i  11 

2.658  7058 
.659  0907 

64.14 
64.17 

2.682  0950 
.682  4900 

65.81 
65.84 

2.706  1014 
.706  5069 

67.57 
67.60 

2-73°  7539 
.731  1705 

69.40 
69.44 

12 

.659  4758 

64.19 

.682  8851 

65.87 

.706  9126 

67-63 

.731  5872 

69.47 

13 

.659  8611 

64.22 

.683  2804 

65.90 

.707  3184 

67.66 

.732  0041 

69.50 

14 

.660  2465 

64.25 

.683  6759 

65.93 

.707  7244 

67.69 

.732  4212 

69-53 

15 

2.660  6320 

64.28 

2.684  0716 

65.96 

2.708  1307 

67.72 

2.732  8385 

69.56 

16 

.661  0178 

64.30 

.684  4674 

65.99 

.708  5371 

67.75 

•733  *559 

69.59 

17 

.661  4037 

64-33 

.684  8634 

66.01 

.708  9436 

67.78 

•733  6736 

69.62 

18 

.661  7897 

64.36 

.685  2596 

66.04 

.709  3504 

67.81 

•734  0914 

69.66 

19 

.662  1760 

64.38 

.685  6559 

66.07 

•709  7573 

67.84 

•734  5°94 

69.69 

20 

2.662  5623 

64.41 

2.686  0524 

66.10 

1.710  1645 

67.87 

2.734  9277 

69.72 

21 

.662  9489 

64.44 

.686  4491 

66.13 

.710  5718 

67.90 

•735  346i 

69.75 

22 
23 

•663  3356 
.663  7225 

64.47 
64.49 

.686  8460 
.687  2430 

66.16 
66.19 

.710  9792 
.711  3869 

67.93 
67.96 

•735  7647 
•736  1835 

69.78 
69.81 

24 

.664  1096 

64.52 

.687  6402 

66.22 

.711  7947 

67.99 

.736  6025 

69.85 

25 

2.664  4968 

64.55 

2.688  0376 

66.25 

2.712  2028 

68.02 

2.737  0216 

69.88 

26 

.664  8842 

64.57 

.688  4352 

66.27 

.712  6110 

68.05 

•737  44io 

69.91 

27 

.665  2717 

64.60 

.688  8329 

66.30 

.713  0194 

68.08 

•737  8605 

69.94 

28 

.665  6594 

64.63 

.689  2308 

66.33 

.713  4279 

68.11 

.738  2803 

69.97 

29 

.666  0473 

64.66 

.689  6289 

66.36 

.713  8367 

68.14 

•738  7002 

70.00 

30 

2.666  4354 

64.69 

2.690  0272 

66.39 

2.714  2456 

68.17 

2.739  I2°3 

70.04 

31 

.666  8236 

64.72 

.690  4256 

66.42 

.714  6547 

68.20 

•739  54o6 

70.07 

32 

.667  2120 

64.74 

.690  8242 

66.45 

.715  0640 

68.23 

•739  961* 

70.10 

33 

.667  6005 

64.77 

.691  2230 

66.48 

.715  4735 

68.26 

.740  3819 

70.13 

34 

.667  9892 

64.80 

.691  6219 

66.51 

.715  8832 

68.29 

.740  8027 

70.16 

35 

2.668  3781 

64.83 

2.692  0210 

66.54 

2.716  2930 

68.32 

2.741  2238 

70.20 

36 

.668  7672 

64.86 

.692  4203 

66.56 

.716  7031 

68.35 

.741  6451 

70.23 

37 

.669  1564 

64.88 

.692  8198 

66.59 

•7«7  "33 

68.38 

.742  0666 

70.26 

38 

.669  5457 

64.91 

.693  2194 

66.62 

.717  5*37 

68.41 

.742  4882 

70.29 

39 

-669  9353 

64.94 

.693  6193 

66.65 

.717  9342 

68.44 

.742  9101 

70.32 

40 

2.670  3250 

64.97 

2.694  0193 

66.68 

2.718  3450 

68.48 

2.743  33*1 

70.36 

41 

.670  7149 

65.00 

.694  4194 

66.71 

.718  7560 

68.51 

•743  7543 

70-39 

42 

.671  1050 

65.02 

.694  8198 

66.74 

.719  1671 

68.54 

•744  1768 

70.42 

43 
44 

.671  4952 
.671  8856 

65.05 
65.08 

.695  2203 
,695  6210 

66.77 
66.80 

.719  5784 
.719  9899 

68.57 
68.60 

.744  5994 
.745  0222 

70.45 
70.48 

45 

2.672  2761 

65.11 

2.696  O2I9 

66.83 

2.720  .4016 

68.63 

*-745  44p 

70.52 

46 

.672  6668 

65.13 

.696  4229 

66.86 

.720  8135 

68.66 

.745  8684 

70.55 

47 

.673  0577 

65.16 

.696  8242 

66.89 

.721  2255 

68.69 

.746  2918 

70.58 

48 

.673  4488 

65.19 

.697  2256 

66.92 

.721  6377 

68.72 

.746  7154 

70.61 

49 

.673  8400 

65.22 

.697  6272 

66.95 

.722  0502 

68.75 

•747  '391 

70.65   . 

5O 

2.674  2314 

65.25 

2.698  0289 

66.97 

2.722  4628 

68.78 

2.747  5631 

70.68 

51 

.674  6230 

65.28 

.698  4308 

67.00 

.722  8756 

68.  81 

•747  9873 

70.71 

52 
i  53 
54 

..675  0147 
.675  4066 
.675  7987 

65.30 
65-33 
65-36 

.698  8330 
.699  2353 
.699  6377 

67.03 
67.06 
67.09 

.723  2885 
.723  7017 
.724  1150 

68.84 
68.88 
68.91 

.748  4116 
.748  8362 
.749  2609 

70.74 
70.78 
70.81 

55 

2.676  1909 

65.39 

2.700  0404 

67.12 

2.724  5286 

68.94 

2.749  6859 

70.84 

56 
57 

.676  5833 
.676  9759 

65.42 
65.44 

.700  4432 
.700  8462 

67.15 
67.18 

.724  9423 
.725  3562 

68.97 
69.00 

.750  mo 
•75°  5364 

70.87 
70.90 

58 
59 

.677  3687 
.677  7616 

6^.47 
65.50 

.701  2494 
.701  6527 

67.21 
67.24 

•7*5  77°3 
.726  1846 

69.03 
69.00 

.750  9619 
.751  3876 

70.94 
70.97 

60 

2.678  1547 

65-53 

2.702  0562 

67.27  |2.726  5990 

69.09 

2.751  8135 

71.00 

599 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

136° 

137° 

138° 

139° 

log  Hi 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

O' 

2.751  8135 

71.00 

2.777  7322 

73.01 

2.804  3895 

75.11 

2.831  8224 

77-32 

1 

•75*  2396 

71.03 

.778  1703 

73-°4 

.804  8403 

75-14 

.832  2864 

77-35 

2 

.752  6659 

71.07 

.778  6087 

73.07 

.805  2912 

75.18 

.832  7506 

77-39 

3 

•753  °925 

71.10 

•779  °472 

73-" 

.805  7424 

75.21 

.833  2151 

77-43 

4 

•753  519* 

7LI3 

•779  4859 

73-J4 

.806  1938 

75-25 

.833  6798 

77-47 

5 

2.753  946i 

71.17 

2.779  9249 

73.18 

2.806  6454 

75.29 

2.834  1447 

77.50 

6 

•754  3732 

71.20 

.780  3641 

73.21 

.807  0973 

75-32 

.834  6098 

77-54 

7 

•754  *°°4 

71.23 

.780  8034 

73-24 

.807  5493 

75-36 

•835  0752 

77-58 

8 

•755  2279 

71.26 

.781  2430 

73.28 

.808  0016 

75.40 

.835  5408 

77.62 

9 

•755  6556 

71.30 

.781  6828 

73-31 

.808  4541 

75-43 

.836  0066 

77.66 

10 

2.756  0835 

71-33 

2.782  1228 

73-35 

2.808  9068 

75-47 

2.836  4727 

77.69 

11 

.756  5116 

71.36 

.782  5630 

73.38 

.809  3597 

75-5° 

.836  9390 

77-73 

12 

•756  9399 

71.40 

.783  0034 

73.42 

.809  8128 

75-54 

•837  4055 

77-77 

13 

•757  3683 

71.43 

.783  4440 

73-45 

.810  2662 

75-58 

.837  8722 

77.81 

14 

•757  797° 

71.46 

.783  8848 

7349 

.810  7197 

75.61 

.838  3392 

77-85 

15 

2.758  2259 

71.49 

2.784  3258 

73-53 

2.811  1735 

75-65 

2.838  8064 

77.89 

16 

.758  6549 

7L53 

.784  7671 

73.56 

.811  6275 

75.69 

.839  2738 

77.92 

17 

.759  0842 

71.56 

.785  2085 

73-59 

.812  0817 

75-72 

.839  7414 

77-96 

18 

•759  5*37 

71-59 

.785  6502 

73-63 

.812  5362 

75.76 

.840  2093 

78.00 

19 

•759  9433 

71.63 

.786  0920 

73.66 

.812  9908 

75-79 

.840  6774 

78.04 

20 

2.760  3732 

71.66 

2.786  5341 

73-7° 

2.813  4457 

75-83 

2.841  1458 

78.08 

21 
22 

.760  8032 
.761  2335 

71.69 

71.73 

.786  9764 
.787  4189 

73-73 
73.76 

.813  9008 
.814  3561 

75-87 
75.90 

.841  6144 
.842  0832 

78.0 
78.15 

23 

.761  6639 

71.76 

.787  8615 

73.80 

.814  8117 

75-94 

.842  5522 

78.19 

24 

.762  0946 

71.79 

.788  3044 

73.83 

.815  2674 

75.98 

.843  0215 

78.23 

25 

2.762  5255 

-71.83 

2.788  7476 

73.87 

2.815  7234 

76.01 

2.843  49°9 

78.27 

26 

.762  9565 

71.86 

.789  1909 

73-9° 

.816  1796 

76.05 

.843  9607 

78.31 

27 

.763  3878 

71.89 

.789  6344 

73-94 

.816  6360 

76.09 

.844  4306 

78-35 

28 

.763  8192 

71.93 

.790  0781 

73-97 

.817  0927 

76.12 

.844  9008 

78.38 

29 

.764  2509 

71.96 

.790  5221 

74.01 

.817  5495 

76.16 

.845  3712 

78.42 

30 

2.764  6827 

71.99 

2.790  9662 

74.04 

2.818  0066 

76.20 

2.845  84*9 

78.46 

31 

.765  1148 

72.03 

.791  4106 

74.08 

.818  4639 

76.23 

.846  3128 

78.50 

32 

.765  5470 

72.06 

.791  8552 

74.11 

.818  9214 

76.27 

.846  7839 

78.54 

33 

•765  9795 

72.09 

.792  3000 

74-'5 

.819  3792 

76.31 

•847  2553 

78.58 

34 

.766  4121 

72.13 

.792  7450 

74.18 

.819  8371 

76.34 

.847  7268 

78.62 

35 

2.766  8450 

72.16 

2.793  19°* 

74.22 

2.820  2953 

76.38 

2.848  1986 

78.66 

36 

.767  2781 

72.19 

.793  6356 

74.25 

.820  7537 

76.42 

.848  6707 

78.69 

37 

.767  7113 

72.23 

.794  0813 

74.29 

.821  2123 

7646 

.849  1430 

78-73 

38 

.768  1448 

72  26 

•794  5271 

74.32 

.821  6712 

76.49 

.849  6155 

78-77 

39 

.768  5784 

72.29 

•794  9731 

74-  36 

.822  1302 

76.53 

.850  0882 

78.81 

40 

2.769  0123 

7^-33 

2-795  4i94 

74.40 

2.822  5895 

76-57 

2.850  5612 

7!-!5 

41 

.769  4464 

72.36 

•795  8659 

74-43 

.823  0491 

76.60 

.851  0344 

78.89 

42 

.769  8806 

72-39 

.796  3126 

74-47 

.823  5088 

76.64 

.851  5079 

78.93 

1  43 

.770  3151 

72.43 

.796  7595 

74.50 

.823  9688 

76.68 

.851  9816 

78.97 

1  44 

.770  7498 

72.46 

.797  2066 

74-54 

.824  4289 

76.72 

•852  4555 

79.01 

45 

2.771  1846 

72.50 

2-797  6539 

74-58 

2.824  8894 

76-75 

2.852  9297 

79.05 

46 

.771  6197 

7^-53 

.798  1015 

74.61 

.825  3500 

76.79 

.853  4041 

79.08 

47 

.772  0550 

72.56 

.798  5492 

74.64 

.825  8108 

7683 

.853  8787 

79.12 

48 

.772  4905 

72.60 

.798  9972 

74.68 

.826  2719 

76.87 

854  3535 

79.16 

49 

.7-2  9262 

72.63 

•799  4454 

74-7  « 

.826  7332 

76.90 

.854  8286 

7920 

:  so 

2.7-3  3621 

72.67 

2-799  8938 

74-75 

2.827  1947 

76.94 

2.855  304o 

79.24 

51 

773  7982 

72.70 

.800  3424 

74-79 

.827  6565 

7698 

•855  7795 

79.28 

52 

.774  2344 

72-73 

.800  7912 

74.82 

.828  1185 

77.01 

•856  2553  i  79.32 

53 

.774  6709 

72.77 

.801  2402 

74.86 

.828  5807 

77-°5 

.856  73x4 

79-3& 

54 

•775  i°77 

72.80 

.801  6895 

74.89 

.829  0431 

77.09 

.857  2077 

79.40 

55 

2.775  5446 

72.84 

2.802  1390 

74-93 

2.829  5058 

77.13 

2.857  6842 

79-44 

56 

•775  98l7 

72.87 

.802  5886 

74-96 

.829  9686 

77.16 

.858  1609 

79.48 

57 

.776  4190 

72.90 

.803  0385 

75.00 

.830  4317 

77.20 

.858  6379 

79.52 

58 
59 

.776  8565 
.777  2942 

72.94 
72.97 

.803  4886 
.803  9390 

75.04 
75.08 

.830  8951 
.831  3586 

77.24 
77.28 

.859  1151 
.859  5926 

79.56 
79.60 

60 

2.777  7322 

73.01 

2.804  3895 

75.11 

2.831  8224 

77-32 

2.860  0703 

79.64 

600 


TABLE  VI. 

]?  or  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

140° 

141° 

142° 

143° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

log  M. 

Diff.  1". 

logM. 

Diff.  1". 

O' 

2.860  0703 

79.64 

2.889  1754 

82.08 

2.919  1831 

84.65 

2.950  1420 

87-37 

1 

.860  5482 

79.68 

.889  6680 

82.12 

.919  6911 

84.70 

.950  6664 

87-41 

2 

.861  0264 

79.72 

.890  1609 

82.16 

.920  1994 

84.74 

.951  1910 

87.46 

3 

.861  5048 

79.76 

.890  6540 

82.20 

.920  7080 

84.78 

•951  7159 

87.50 

4 

.861  9835 

79.80 

.891  1473 

82.25 

.921  2169 

84.83 

.952  2411 

87-55 

5 

2.862  4624 

79.84 

2.891  6409 

82.29 

2.921  7260 

84.87 

2.952  7665 

87.60 

6 

.862  9415 

79.88 

.892  1348 

82.33 

.922  2353 

84.92 

•953  29*3 

87.65 

7 

.863  4209 

79.92 

.892  6289 

82.37 

.922  7450 

84.96 

•953  8183 

87.69 

8 

.863  9005 

79.96 

•893  I233 

82.41 

.923  2549 

85.01 

.954  3446 

87.74 

9 

.864  3803 

80.00 

•893  6l79 

82.46 

.923  7650 

85.05 

•954  8711 

87.79 

10 

2.864  8604 

80.04 

2.894  1127 

82.50 

2.924  2755 

85.10 

2.955  398o 

87.81 

11 

.865  3408 

80.08 

.894  6078 

82.54 

.924  7862 

85.14 

•955  9*51 

87.88 

12 

.865  8213 

80.12 

.895  1032 

82.58 

.925  2972 

85.18 

.956  4525 

87.93 

13 

.866  3021 

80.16 

.895  5989 

82.63 

.925  8084 

85.23 

.956  9802 

87.97 

14 

.866  7832 

80.20 

.896  0948 

82.67 

.926  3199 

85.27 

-957  5°8^ 

88.02 

15 

2.867  2645 

80.34 

2.896  5909 

82.71 

2.926  8317 

85.32 

2.958  0365 

88.07 

16 

.867  7460 

80.28 

.897  0873 

82.75 

.927  3437 

85.36 

.958  5651 

88.11 

17 

.868  2278 

80.32 

.897  5839 

82.79 

.927  8560 

85.41 

•959  °939 

88.16 

18 

.868  7098 

80.36 

.898  0808 

82.84 

.928  3686 

85.45 

.959  6230 

88.21 

19 

.869  1921 

80.40 

.898  5780 

82.88 

.928  8814 

85.50 

.960  1524 

88.26 

20 

2.869  6746 

80.44 

2.899  0754 

82.92 

2.929  3945 

85.54 

2.960  6821 

88.30 

21 

.870  1573 

80.48 

.899  5730 

82.96 

.929  9079 

85.59 

.961  2I2O 

88-35 

22 

.870  6403 

80.52 

.900  0709 

83.01 

.930  4216 

85.61 

.961  7423 

88.40 

23 

.871  1235 

80.56 

.900  5691 

83.05 

•93°  9355 

85.68 

.962  2728 

88.45 

24 

.871  6070 

80.60 

.901  0675 

83.09 

.931  4497 

85.72 

.962  7036 

88.49 

25 

2.872  0907 

80.64 

2.901  5662 

83.13 

2.931  9641 

85.77 

*-963  3347 

88.54 

26 

.872  5747 

80.68 

.902  0651 

83.18 

.932  4788 

85.81 

.963  8661 

88  59 

27 

.873  0589 

80.72 

.902  5643 

83.22 

.932  9938 

85.86 

.964  3978 

88.64 

28 

•873  5433 

80.76 

.903  0638 

83.26 

•933  S^1 

85.91 

.964  9297 

88.68 

29 

.874  0280 

80.80 

.903  5635 

83.31 

•934  °M7 

85.95 

.965  4620 

88.73 

30 

2.874  5129 

80.84 

2.904  0635 

83-35 

2.934  5405 

85.99 

2.965  9945 

88.78 

31 

.874  9981 

80.88 

.904  5637 

83-39 

•935  °565 

86.04 

.966  5273 

88.83 

32 

•875  4835 

80.92 

.905  0642 

83-43 

•935  57^9 

86.08 

.967  0604 

88.87 

33 

.875  9692 

80.96 

.905  5649 

83.48 

.936  0895 

86.13 

.967  5938 

88.92 

34 

.876  4551 

81.01 

.906  0659 

83.52 

.936  6064 

86.17 

.968  1275 

88.97 

35 
36 

2.876  9413 

.877  4277 

81.05 
81.09 

2.906  5672 
.907  0687 

83.56 
83.61 

2.937  1236 
•937  6410 

86.22 

86.26 

2.968  6615 
.969  1957 

89.02 
89.07 

37 

.877  9143 

81.13 

.907  5704 

83.65 

.938  1587 

86.31 

.969  7303 

89.12 

38 

.878  4012 

81.17 

.908  0725 

83.69 

.938  6767 

86.35 

.970  2651 

89.17 

39 

.878  8883 

81.21 

.908  5748 

83.74 

.939  1950 

86.40 

.970  8002 

89.21 

40 

*-879  3757 

81.25 

2.909  0773 

83.78 

2.939  7135 

86.45 

2.971  3356 

89.26 

41 

.879  8633 

81.29 

.909  5801 

83.82 

.940  2323 

86.49 

•971  8713 

89.31 

42 

.880  3512 

81.33 

.910  0832 

83.87 

.940  7514 

86.54 

.972  4073 

89.36 

43 

.880  8393 

81.37 

.910  5865 

83.91 

.941  2708 

86.58 

.972  9436 

89.40 

44 

.881  3277 

81.42 

.911  0901 

83-95 

.941  7904 

86.63 

•973  4801 

89.45 

45 

2.881  8163 

81.46 

2.911  5940 

8399 

2.942  1103 

86.67 

2.974  0170 

89.50 

46 

.882  3052 

81.50 

.912  0981 

84.04 

.942  8305 

86.72 

•974  5541 

89.55 

47 

.882  7943 

81.54 

.912  6024 

84.08 

•943  35io 

86.77 

•975  °9l6 

89.60 

48 

.883  2837 

81.58 

.913  1070 

84.13 

•943  8717 

86.81 

•975  6293 

89.65 

49 

•883  7733 

81.62 

.913  6119 

84.17 

•944  3927 

86.86 

.976  1673 

89.69 

50 

2.884  2631 

81.66 

2.914  1171 

84.22 

2.944  9140 

86.90 

2.976  7056 

89.74 

51 

.884  7532 

81.70 

.914  6225 

84.26 

•945  4355 

86.95 

.977  2442 

89.79 

52 
53 

.885  2436 
.885  7342 

81.75 
81.79 

.915  1282 
.915  6341 

84.30 
84-34 

•945  9574 
.946  4795 

87.00 
87.04 

•977  7831 
.978  3223 

a  a  £  o 

89.84 
89.89 

54 

.886  2251 

81.83 

.916  1403 

84.39 

•947  OOI9 

87.09 

.978  8618 

89.94 

55 

2.886  7162 

81.87 

2.916  6468 

84.43 

2.947  5245 

87-13 

1.979  4015 

89.99 

56 
57 

58 
59 

.887  2075 
.887  6991 
.888  1910 
.888  6831 

81.91 

81.95 
81.99 
82.04 

.917  1535 
.917  6605 
.918  1678 
.918  6753 

84.48 
84.52 
84.56 
84.61 

.948  0475 
.948  5707 
.949  0942 
•949  6180 

87.18 
87.23 
87.27 
87.3z 

•979  94i6 
.980  4820 
.981  1226 
.981  6636 

90.01 
90.08 
90.13 
90.18 

60 

2.889  1754 

82.08 

2.919  1831 

84.65 

2.950  1420 

87-37 

2.982  1048 

90.*^ 

601 


TABLE  VI. 

Far  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

144° 

145° 

146° 

147° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

log  M. 

Diff.  1". 

log  M. 

Diff.  1". 

0' 

2.982  1048 

90.23 

3.015  1281 

93.26 

3.049  2733 

96.47 

3.084  6070 

99.87 

1 

.982  6463 

90.28 

.015  6878 

93-31 

.049  8522 

96.52 

.085  2064 

99.92 

2 

.983  1882 

9°-33 

.016  2478 

93.36 

.050  4315 

96.58 

.085  8061 

99.98 

3 

.983  7303 

qo.3S 

.016  8082 

93.42 

.051  on  2 

96-63 

.086  4062 

100.04 

4 

.984  2727 

90.43 

.017  3688 

93-47 

.051  5911 

96.69 

.087  0066 

100.10  1 

5 

2.984  8154 

90.48 

3.017  9298 

93.52 

3.052  1714 

96.74 

3.087  6073 

ioo.  16 

6 

.985  3584 

9°-53 

.018  4911 

93-57 

.052  7520 

96.80 

.088  2085 

100.22   ' 

7 

.985  9017 

90.58 

.019  0526 

93.62 

.053  3329 

96.85 

.088  8099 

100.28 

8 

.986  4453 

90.63 

.019  6145 

93.68 

.053  9142 

96.91 

.089  4118 

100.33 

9 

.986  9892 

90.67 

.020  1768 

93-73 

•°54  4959 

96.96 

.090  0140 

100.39 

1O 

i-987  5334 

90.72 

3.020  7393 

93-78 

3.055  0778 

97.01 

3.090  6165 

100.45 

11 

.988  0779 

90.77 

.021  3021 

93.83 

.055  6601 

97.07 

.091  2194 

100.51 

12 

.988  6227 

90.82 

.O2I  8653 

93.89 

.056  2427 

97.I3 

.091  8226 

100.57 

13 

.989  1678 

90.87 

.022  4288 

93-94 

.056  8256 

97.19 

.092  4262 

100.63 

14 

.989  7132 

90.92 

.022  9926 

93-99 

.057  4089 

97.24 

.093  0302 

100.69 

15 

2.990  2589 

90.97 

3.023  5567 

94.04 

3.057  9925 

97.30 

3.093  6345 

100.75 

16 

.990  8049 

91.02 

.024  121  I 

94.10 

.0.58  5765 

97-35 

.094  2392 

I00.8I 

17 

.991  3512 

91.07 

.024  6859 

94.15 

.059  1608 

97.41 

.094  8442 

100.87 

18 

.991  8977 

91.12 

.025  2509 

94.20 

.059  7454 

97-47 

.095  4496 

100.93 

19 

.992  4446 

91.17 

.025  8163 

94.26 

.060  3304 

97.52 

.096  0553 

100.98 

20 

•A.  992  9918 

91.22 

3.026  3820 

94-  3  1 

3.060  9157 

97.58 

3.096  6614 

101.04 

21 

•993  5393 

91.27 

.026  9480 

94-36 

.061  5013 

97.63 

.097  2678 

IOI.IO 

22 

.994  0871 

91.32 

.027  5143 

94.41 

.062  0873 

97.69 

.097  8746 

101.16 

23 

•994  6351 

9'-37 

.O28  o8lO 

94-47 

.062  6736 

97-75 

.098  4818 

101.22 

24 

•995  I&35 

91.42 

.028  6479 

94.52 

.063  2602 

97.80 

.099  0893 

101.28 

25 

1-995  73" 

91.47 

3.029  2152 

94-57 

3.063  8472 

97.86 

3.099  6972 

101.34 

26 

.996  2812 

91.52 

.029  7828 

94-63 

.064  4345 

97.91 

.100  3054 

101.40 

27 

.996  8305 

91-57 

.030  3507 

94.68 

.065  O222 

97-97 

.100  9140 

101.46 

28 

•997  3801 

91.62 

.030  9190 

94-73 

.065  6101 

98.03 

.101  5230 

101.52 

29 

•997  93°° 

91.67 

.031  4875 

94-79 

.066  1985 

98.08 

.102  1323 

101.58 

30 
31 

2.998  4802 
•999  °3°7 

91.72 
91.77 

3.032  0564 
.032  6256 

94.84 
94.89 

3.066  7872 
.067  3762 

98.14 

98.20 

3.102  7420 
.103  352O 

101.64 
101.70 

32 

•999  58l5 

91.82 

.033  1951 

94-94 

.067  9655 

98.25 

.103  9624 

101.76 

33 

3.000  1326 

91.87 

.033  7650 

95.00 

.068  5552 

98.31 

.104  5732 

101.82 

34 

.000  6840 

91.93 

•034  3351 

95.05 

.069  1453 

98.37 

.105  1843 

101.88 

35 

3.001  2357 

91.98 

3.034  9056 

95.11 

3.069  7357 

98.42 

3-105  7958 

101.94 

36 

.001  7877 

92.03 

.035  4764 

95.16 

.070  3264 

98.48 

.106  4076 

102.00 

37 

.002  3400 

92.08 

.036  0475 

95.22 

.070  9174 

98.54 

.107  0198 

IO2.O7 

38 

.002  8926 

92.13 

.036  6190 

95.27 

.071  5088 

98.60 

.107  6324 

102.13 

39 

.003  4456 

92.18 

.037  1908 

95-32 

.072  1006 

98.65 

.108  2454 

102.19 

40 

3.003  9988 

92.23 

3.037  7629 

95.38 

3.072  6927 

98.71 

3.108  8587 

102.25 

41 

.004  5523 

92.28 

•°38  3353 

95-43 

.073  2851 

98.77 

.109  4723 

102.31 

42 

.005  1062 

9L33 

.038  9080 

95.48 

.073  8779 

98.82 

.no  0864 

102.37 

43 

.005  6603 

92.38 

.039  4811 

95-54 

.074  4710 

98.88 

.no  7008 

102.43 

44 

.006  2148 

92.44 

.040  0545 

95.60 

.075  0645 

98.94 

•"'  3*55 

102.49 

45 

3.006  7696 

92.49 

3.040  6282 

95-65 

3.075  6583 

99.00 

3.111  9306 

102-55 

46 

47 

.007  3246 
.007  8800 

92.54 
92.59 

.041  2023 
.041  7767 

95.70 
95.76 

.07*  2524 
.076  8469 

99.05 
99.11 

.112  5461 
.113  l620 

102.61 
102.67 

48 

.ooS  4357 

92.64 

.042  3514 

95.81 

.077  4418 

99.17 

.113  7782 

102.73 

49 

.008  9917 

92.69 

.042  9264 

95.86 

.078  0370 

99-13 

.114  3948 

102.80 

50 

3.009  5480 

92.74 

3.043  5017 

95-91 

3.078  6325 

99.28 

3.115  0118 

102.86 

51 

.010  1046 

92.79 

.044  0774 

95-97 

.079  2284 

99-34 

.115  6291 

102.92 

52 

.010  6615 

92.85 

.044  6534 

96.01 

.079  8246 

99.40 

.116  2468 

102.98 

53 

.on  2188 

92.90 

.045  2297 

96.08 

.080  4212 

99.46 

.116  8649 

103.04 

54 

.011  7763 

92.95 

.045  8064 

96.14 

.081  0181 

99.52 

.117  4833 

103.10 

55 

3.012  3342 

93.00 

3.046  3834 

96.19 

3.081  6154 

99-57 

3.118  1022 

103.16 

56 

.012  8923 

93-05 

.046  9607 

96.25 

.082  2130 

99.63 

.118  7213 

103.23 

57 

.013  4508 

93.10 

•°47  5383 

96.30 

.082  8no 

99.69 

.119  3409 

103.29 

58 

.014  0096 

93.16 

.048  1163 

96.36 

.083  4093 

99-75 

.119  9608 

103.35 

59 

.014  5687 

93.21 

.048  6946 

96.41 

.084  0080 

99.81 

.120  5811 

103.41 

60 

3.015  I28l 

93.26 

3.049  2733 

96.47 

3.084  6070 

99.87 

3.I2I  20l8 

103.48 

602 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

148° 

149° 

150° 

151° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0' 

.121  20l8 

103.48 

.159  1767 

107.31 

.198  4984 

111.41 

.239  3820 

115.77 

1 

.121  8228 

103.54 

.159  7808 

107.38 

.199  1671 

111.48 

.240  0768 

115.85 

2 
3 

.122  4442 
.127  OOOO 

103.60 
103.66 

.160  4253 
.161  0702 

107.45 
107.51 

.199  8361 

.200  5056 

"X-SS 
111.62 

.240  7722 
.241  4680 

115.92 
116.00 

4 

.123  6882 

103.72 

.161  7154 

107.58 

.201  1755 

111.69 

.242  1642 

116.08 

5 

.124  3107 

103.79 

.162  3611 

107.65 

.201  8459 

111.76 

3.242  8608 

116.15 

6 

.124  9336 

103.85 

.163  0072 

107.71 

.202  5166 

111.83 

.243  5580 

116.23 

7 

.125  5569 

103.91 

.163  6536 

107.78 

.203  1878 

111.90 

.244  2556 

Il6.70 

8 
9 

.126  1805 
.126  8045 

103.97 
104.04 

.164  3005 
.164  9478 

107.85 
107.91 

.203  8594 
.204  5315 

111.97 

112.04 

.244  9536 
.245  6521 

116.38 
116.45 

10 
11 
12 

.127  4289 
.128  0537 
.128  6789 

104.10 
104.16 
104.22 

3-i65  5955 
.166  2435 
.166  8920 

107.98 
108.04 
108.11 

.205  2040 
.205  8769 
.206  5502 

112.  II  • 

112.18 
112.26 

3.246  3511 

.247  0505 
.247  7503 

116.53 

116.61 
116.68 

13 

.129  3044 

104.29 

.167  5409 

108.18 

.207  2239 

112.33 

.248  4507 

116.76 

14 

.129  9303 

104.35 

.168  1901 

108.25 

.207  8981 

1  1  2.40 

.249  1515 

116.84 

15 

.130  5566 

104.41 

3.168  8398 

108.31 

3.208  5727 

112.47 

3.249  8527 

116.91 

16 

.131  1833 

104.48 

.169  4899 

108.38 

.209  2478 

112.54 

.250  5544 

116.99 

17 

.131  8103 

104.54 

.170  1404 

108.45 

.209  9232 

112.61 

.251  2566 

117.07 

18 
19 

.132  4377 
•133  °655 

104.60 
104.67 

.170  7913 
.171  4426 

108.51 
108.58 

.210  5991 
.211  2755 

112.69 
112.76 

.251  9592 
.252  6623 

117.14 

117.22 

20 
21 

3.133  6937 
.134  3223 

104.73 
104.79 

3.172  0942 
.172  7463 

108.65 

108.72 

3-2II  9522 
.212  6294 

112.83 
112.90 

3.253  3658 
.254  0698 

117.30 
"7-37 

22 

.134  9512 

104.86 

.173  3988 

108.78 

.213  3070 

112.97 

•*54  7743 

117.45 

23 
24 

•135  5«o5 

.136  2IO2 

104.92 
104.98 

.174  0517 
.174  7051 

108.85 
108.92 

.213  9851 
.214  6636 

113.05 

113.12 

.255  4792 
.256  1846 

117.53 
117.60 

25 

3.136  8403 

105.05 

3-J75  3588 

108.99 

3.215  3425 

113.19 

3.256  8905 

117.68 

26 

.137  4708 

105.11 

.176  0129 

109.06 

.2l6  0219 

113.26 

.257  5968 

117.76 

27 

.138  1016 

105.17 

.176  6674 

109.12 

.2l6  7017 

"3-34 

.258  3036 

117.84 

28 
29 

.138  7329 
.139  3645 

105.24 
105.30 

.177  3224 
.177  9777 

109.19 
109.26 

.217  3819 

.218  0626 

113.41 
113-48 

.259  0109 
.259  7186 

117.91 

117.99 

30 

3.139  9965 

105.36 

S-^8  6335 

109.33 

3.218  7437 

"3-55 

3.260  4268 

118.07 

31 

.140  6289 

i°5-43 

.179  2897 

109.40 

.219  4252 

113.63 

.261  1354 

118.15 

32 

.141  2616 

105.49 

.179  9462 

109.46 

.220  1072 

113.70 

.261  8446 

118.23 

33 
34 

.141  8948 
.142  5283 

105.55 
105.62 

.180  6032 
.181  2606 

i°9-53 
109.60 

.220  7896 
.221  4724 

"3-77 
113.84 

.262  5542 
.263  2642 

118.30 
118.38 

35 

3.143  1622 

105.68 

3.181  9184 

109.67 

3.222  1557 

113.92 

3.263  9747 

118.46 

36 

•143  7965 

105.75 

.182  5766 

109.74 

.222  8395 

113.99 

.264  6857 

"8.54 

37 
38 

.144  4312 
.145  0663 

105.81 
105.87 

•183  2353 
.183  8943 

109.81 
109.87 

.223  5276 
.224  2082 

114.06 
114.14 

265  3972 
.266  1091 

118.62 
118.70 

39 

.145  7018 

105.94 

.184  5538 

109.94 

.224  8933 

114.21 

.266  8216 

118.77 

40 
41 

3.146  3376 
.146  9739 

106.00 
106.07 

3.185  2136 
.185  8739 

IIO.OI 

110.08 

3.225  5788 
.226  2647 

114.28 
114.36 

3-^7  5345 
.268  2478 

118.85 
118.93 

42 

.147  6105 

106.14 

.186  5346 

110.15 

.226  9511 

"4-43 

.268  9616 

119.01 

43 

.148  2475 

106.20 

.187  1957 

110.22 

.227  6379 

114.51 

.269  6759 

119.09 

44 

.148  8849 

106.27 

.187  8572 

110.29 

.228  3252 

114.58 

.270  3907 

119.17 

45 

3.149  5227 

106.33 

3.188  5192 

110.36 

3.229  0129 

114.65 

3.271  1060 

119.25 

46 

.150  1609 

106.40 

.189  1815 

110.43 

.229  7010 

114.73 

.271  8217 

"9-33 

47 

.150  7995 

1  06.4(1 

.189  8443 

110.50 

.230  3896 

114.80 

.272  5379 

119.41 

48 

.151  4385 

106.53 

.190  5075 

110.57 

.231  0786 

114.88 

.273  2546 

119.49 

49 

.152  0778 

106.59 

.191  1711 

110.64 

.231  7681 

114.95 

.273  9717 

119.57 

50 

3.152  7176 

106.66 

3.191  8351 

110.71 

3.232  4581 

115.03 

3.274  6894 

119.65 

51 
52 

•153  3577 
.153  9983 

106.72 
106.79 

.192  4996 
.193  1644 

110.77 
110.84 

.233  1484 
.233  8392 

115.10 
115.17 

.275  4075 
.276  1261 

"9-73 
119.81 

53 

.154  6392 

106.85 

.193  8297 

110.91 

•»34  53°5 

115.25 

.276  84.52 

119.89 

54 

.155  2805 

106.92 

.194  4954 

110.98 

.235  2222 

"5-3* 

.277  5647 

119.97 

55 
56 
57 

3.155  9222 
.156  5643 
.157  2068 

106.99 

107.05 
107.12 

3.195  1615 
.195  8281 
.196  4950 

111.05 
III.  12 
III.I9 

3.235  9144 
.236  6070 
.237  3OOI 

115.40 
115.47 
"5-55 

3.278  2848 
.279  0053 
.279  7263 

120.05 
120.13 

120.21 

58 
59 

.157  8497 
.158  4930 

107.18 
107.25 

.197  1624 
.197  8302 

III.26 
111.34 

.237  9936 
.238  6876 

115.62 
115.70 

.280  4477 
.281  1697 

120.29 
120-37 

GO 

3-i<;Q  1367 

107.31 

3.198  4984 

III.4I 

3.239  3820 

115.77 

3.281  8921 

120.45 

603 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

152° 

153° 

154° 

155° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

DiF  1". 

logM. 

Diff.  1". 

O' 

3.281  8921 

120.45 

3.326  1448 

125.46 

3.372  2684 

130.85 

3.420  4064 

136.66 

1 

.282  6151 

120.53 

.326  8978 

1^5-55 

•373  °538 

130.94 

.421  2266 

136.76 

2 
3 

.283  3385 
.284  0624 

120.61 
120.69 

.327  6513 
.328  4054 

125.63 
125.72 

•373  8397 
.374  6262 

131.04 
131.13 

.422  0475 
.422  8690 

136.86 
136.96 

4 

.284  7868 

120.77 

.329  1600 

125.81 

•375  4133 

131.22 

.423  6910 

137.06 

5 

3.285  5116 

120.85 

3.329  9151 

125.89 

3.376  2009 

131.32 

3.424  5137 

137.16 

6 

.286  2370 

120.93 

.330  6707 

125.98 

.376  9890 

131.41 

•425  3370 

137.26 

7 

.286  9628 

121.01 

.331  4268 

126.07 

•377  7778 

131.50 

.426  1609 

'37-37  ; 

8 

.287  6891 

121.  10 

.332  1835 

126.16 

•378  567i 

131.60 

.426  9854 

J37-47  | 

9 

.288  4160 

I2I.I8 

.332  9407 

126.24 

•379  3570 

131.69 

.427  8105 

137-57 

1O 
11 

3.289  1433 
.289  8711 

121  26 
121.34 

3-333  6984 
•334  4567 

126.33 
126.42 

3.380  1474 
.380  9384 

131.79 
131.88 

3.428  6362 
.429  4626 

137.67 

137.77 

12 

.290  5993 

121.42 

•335  2154 

126.51 

.381  7300 

131.98 

.430  2895 

137.88 

13 

.291  3281 

121.50 

•335  9747 

126.59 

.382  5221 

132.07 

.431  1171 

I37-98 

14 

.292  0574 

121.59 

•336  7346 

126.68 

•383  3H8 

132.16 

.431  9452 

138.08 

15 

3292  7872 

121.67 

3-337  4949 

126.77 

3.384  1081 

132.26 

3.432  7740 

138.18 

16 

•293  5I74 

121.75 

.338  2558 

126.86 

.384  9019 

132.35 

•433  6034 

138.29 

17 

.294  2481 

121.83 

•339  OI72 

126.95 

•385  6963 

132.45 

•434  4334 

'38.39 

18 

.294  9794 

121.91 

•339  7792 

127.03 

.386  4913 

132.54 

.435  2641 

138.49 

19 

.295  7111 

122.00 

•34°  5417 

127.12 

.387  2869 

132.64 

•436  °953 

138.59 

20 

3.296  4433 

122.08 

3-341  3047 

127.21 

3.388  0830 

132.73 

3.436  9272 

138.70 

21 

.297  1761 

I22.I6 

.342  0682 

127.30 

.388  8797 

132.83 

•437  7597 

138.80  ! 

22 

.297  9093 

122.24 

.342  8323 

127-39 

.389  6770 

J32-93 

.438  5928 

138.90  ; 

23 

.298  6430 

122.33 

•343  5969 

127.48 

-39°  4749 

133.02 

.439  4266 

139.01 

24 

.299  3772 

122.41 

•344  3620 

127.57 

.391  2733 

133.12 

.440  2609 

139.11 

25 

3.300  1119 

122.49 

3-345  I277 

127.66 

3.392  0723 

133.22 

3.441  0959 

139.22 

26 

.300  8471 

122.58 

•345  8939 

'27-75 

.392  8719 

'33-31 

.441  9315 

139.32 

27 

.301  5828 

122.66 

.346  6606 

127.84 

•393  672o 

133.41 

.442  7677 

13942 

28 

.302  3190 

122.74 

•347  4279 

127-93 

•394  4728 

'33-5° 

.443  6046 

'39-53 

29 

.303  0557 

122.83 

.348  1958 

128.02 

•395  2741 

133.60 

.444  4421 

139-63 

30 

3.303  7929 

122.91 

3.348  9641 

128.11 

3.396  0760 

133.70 

3.445  2802 

139-74 

31 

.304  5306 

122.99 

•349  733° 

128.19 

.396  8785 

J33-79 

.446  1189 

139.84 

32 

.305  2688 

123.08 

•35°  5°24 

128.28 

•397  6815 

133-89 

•446  9583 

139-95  ! 

33 

.306  0075 

123.16 

.351  2724 

128.37 

.398  4852 

1  33-99 

•447  7983 

140.05 

34 

.306  7468 

123.24 

.352  0429 

128.46 

•399  2894 

134.09 

.448  6389 

140.16 

35 

3.307  4865 

123.33 

3.352  8140 

128.55 

3.400  0942 

134.19 

3.449  4802 

140.26 

36 

.308  2267 

123.41 

•353  5856 

128.65 

.400  8996 

134.28 

•45°  3221 

140.37 

37 
38 

.308  9674 
.309  7086 

123.50 

123.58 

•354  3577 
•355  '3°4 

128.74 
128.83 

.401  7056 
.402  5122 

134-38 

134.48 

.451  1646 
•452  0077 

140.47 
14057 

39 

.310  4504 

123.66 

•355  9°37 

128.92 

.403  3193 

134-57 

.452  8515 

140.68 

40 

3.311  1926 

123.75 

3.356  6774 

129.01 

3.404  1270 

134.67 

3-453  6959 

140.79 

41 

•311  9354 

123.83 

•357  45i7 

129.10 

.404  9354 

134-77 

•454  541° 

140.90 

42 

.312  6786 

123.92 

.358  2266 

129.19 

•4°5  7443 

I34-87 

•455  3867 

141.00 

43 

•3*3  4224 

124.00 

.359  0020 

129.28 

.406  5538 

134-97 

.456  2330 

141.11 

44 

.314  1667 

124.09 

•359  778o 

129.37 

.407  3639 

I35-07 

.457  0800 

141.21 

45 

3-3H  9H5 

124.17 

3.360  5545 

129.46 

3.408  1746 

135.16 

3.457  9276 

141.32 

46 

.315  6567 

124.26 

.361  3316 

129.56 

.408  9859 

135.26 

•458  7759 

141.43  i 

47 

.316  4025 

124.34 

.362  1092 

129.65 

.409  7977 

'35-36 

.459  6248 

141.54  ! 

48   -317  1489 

1  24-43 

.362  8873 

129.74 

.410  6102 

I35-46 

.460  4743 

141.64 

49  -31?  8957 

124.51 

.363  6660 

129.83 

.411  4233 

I35-56 

.461  3245 

141.75 

50  3-3  l8  643° 

124.60 

3-364  4453 

129.92 

3.412  2369 

135.66 

3.462  1753 

141.86 

51 

.319  3909 

124.68 

.365  2251 

130.01 

.413  0512 

135.76 

.463  0268 

141.97 

52 

.320  1392 

124.77 

.366  0055 

130.11 

.413  8660 

135.86 

.463  8789 

142.07 

53 

.320  8881 

124.86 

.366  7864 

130.20 

.414  6815 

135.96 

.464  7317 

I42.I8 

54 

.321  6375 

124.94 

.367  5679 

130.29 

•4*5  4975 

136.06 

•465  5851 

142.29 

55 

3.322  3874 

125.03 

3.368  3499 

130.38 

3.416  3142 

136.16 

3.466  4392 

142.40 

56 

.323  1379 

125.11 

.369  1325 

130.48 

.417  1314 

136.26 

•467  2939 

142.51 

57 

.323  8888 

125.20 

.369  9156 

*3°-57 

.417  9492 

136.36 

.468  1492 

I42.6l 

58 

.324  6403 

125.29 

.370  6993 

130.66 

.418  7677 

136.46 

.469  0052 

142.72 

59 

.325  3923 

125-37 

.371  4836 

130.76 

.419  5867 

136.56 

.469  8619 

142.83 

60 

3.326  1448 

125.46 

3.372  2684 

130.85 

3.420  4064 

136.66 

3.470  7192 

142.94 

604 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

156° 

157° 

158° 

159° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0' 

3.470  7192 

142.94 

3.523  3875 

'49-75 

3-578  6154 

157.17 

3.636  6351 

165.28 

1 
2 
3 

•47'  5772 
.472  4358 

•473  2951 

'43-°5 
143.16 
143.27 

.524  2864 
.525  1860 
.526  0863 

149.87 
149.99 
150.11 

•579  5588 
.580  5030 
.581  4480 

157.30 

'57-43 
'57-56 

.637  6274 
-638  6202 
.639  6140 

'6542 
165.56 

4 

.474  1550 

I43-38 

.526  9873 

150.23 

•S8^  3937 

157.69 

.640  6087 

165.85 

5 
6 

3.475  0156 
•475  8769 

143.49 
143.60 

3.527  8890 
.528  7915 

'50-35 
150.47 

3.583  3403 
.584  2876 

157.82 
'57-95 

3.641  6042 
.642  6006 

165.99 
166.13 

7 

.476  7388 

'43-71 

.529  6947 

150.59 

•585  2357 

158.08 

.643  5978 

166.28 

8 

.477  6014 

143-82 

.530  5985 

150.71 

.586  1846 

158.21 

.644  5959 

166.42 

9 

.478  4646 

'43-93 

•53'  5°3' 

150-83 

•587  '342 

'58-34 

•645  5948 

166.56 

10 

3.479  3285 

144.04 

3-532  4085 

150.95 

3.588  0847 

158.47 

3.646  5946 

166.71 

11 
12 

.480  1931 
.481  0583 

144-15 

144.26 

•533  3'45 
.534  2213 

151.07 
151.19 

•589  0359 
.589  9880 

158.61 
158.74 

•647  5953 
.648  5968 

166.85 
166.99 

13 

.481  9242 

'44-37 

.535  1288 

151.31 

.590  9408 

158.87 

.649  5992 

167.14 

14 

.482  7907 

144-48 

•536  0370 

151.43 

.591  8944 

159.00 

.650  6025 

167.28 

15 

3.483  6579 

'44-59 

3.536  9459 

'5'-55 

3.592  8488 

'59-'3 

3.651  6066 

167.42 

16 

.484  5258 

144.70 

•537  8556 

151.67 

•593  8040 

159.26 

.652  6116 

167.57 

17 

•485  3944 

144-81 

.538  7660 

151.79 

•594  7600 

159.40 

.653  6175 

167.72 

18 
19 

.486  2636 
•487  '335 

'44-93 

145.04 

•539  6771 
.540  5890 

151.91 

152.04 

•595  7167 
•596  6743 

'59-53 
159.66 

.654  6242 
•655  6318 

167.86 
168.01 

20 
21 
22 

3.488  0040 
.488  8752 
•489  7472 

145.15 
145.26 

'45-37 

3-54'  5°'5 
.542  4148 

•543  3289 

152.16 
152.28 
152.40 

3-597  6327 
.598  5919 
•599  55'8 

'59-79 
'59-93 
160.06 

[.656  6403 
•657  6497 
.658  6599 

168.15 
168.30 
168.45 

23 

.490  6198 

'45-49 

.544  2436 

152.52 

.600  5126 

160.19 

•659  6710 

168.59 

24 

•49'  493° 

145.60 

•545  '59' 

152.65 

.601  4742 

160.33 

.660  6830 

168.74 

25 

3.492  3670 

145.71 

3.546  0754 

152.77 

3.602  4365 

160.46 

;.66i  6959 

168.89 

26 

•493  2416 

145-82 

.546  9924 

152.89 

.603  3997 

160.60 

.662  7096 

169.03 

27 

.494  1  1  68 

145-94 

.547  9101 

153.01 

.604  3637 

160.73 

.663  7243 

169.18 

28 

.494  9928 

146.05 

.548  8285 

153-14 

.605  3285 

160.87 

.664  7398 

169.33 

29 

•495  8695 

146.16 

•549  7477 

153.26 

.606  2941 

161.00 

.665  7562 

169.48 

30 

.496  7468 

146.28 

3.550  6677 

'53.38 

1-607  2605 

161.14 

3.666  7735 

169.62 

31 

.497  6248 

146.39 

.551  5883 

'53-5' 

.608  2277 

161.27 

.667  7917 

169.77 

32 

.498  5035 

146.50 

•552  5°97 

153-63 

•609  1957 

161.41 

.668  8108 

169.92 

33 

•499  3828 

146.62 

•553  43'9 

'53-75 

.610  1646 

161.54 

.669  8308 

170.07 

34 

.500  2629 

146.73 

•554  3548 

153.88 

.611  1342 

161.68 

.670  8516 

170.22 

35 

.501  1436 

146.85 

•555  2785 

154-00 

1-612  1047 

161.81 

.671  8734 

170.37 

36 

.502  0250 

146.96 

.556  2029 

'54-'3 

.613  0760 

161.95 

.672  8961 

170.52 

37 

38 

.502  9071 
.503  7899 

147.08 
147.19 

.557  1280 
•558  0539 

154-25 

.614  0481 

.615  0210 

162.09 
162.22 

.673  9196 
.674  9441 

170.67 
170.82 

39 

.504  6734 

147-3' 

.558  9806 

154.50 

.615  9948 

162.36 

.675  9694 

170.97 

40 

.505  5576 

147-42 

-559  9°8o 

154-63 

.6l6  9693 

162.50 

.676  9957 

171.12 

41 

.506  4425 

'47-54 

.560  8361 

'54-75 

.617  9447 

162.63 

.678  0228 

171.27 

42 

.507  3280 

'47-65 

.561  7650 

154.88 

.618  9209 

162.77 

.679  0509 

171.42 

43 

.508  2143 

147.77 

.562  6947 

155.01 

.619  8980 

162.91 

.680  0799 

'71-57 

44 

.509  IOI2 

147-88 

.563  6251 

'55-'3 

.620  8758 

163.05 

.681  1098 

171.72 

45 

.509  9889 

148.00 

.564  5562 

155.26 

.621  8545 

163.18 

.682  1406 

171.87 

46 

.510  8772 

148.11 

.565  4882 

155.38 

.622  8340 

163.32 

.683  1723 

172.03 

I  47 

.511  7662 

148.23 

.566  4209 

'55-5' 

.623  8144 

163.46 

.684  2049 

172.18  - 

48 
49 

.512  6560 

•5  '3  5464 

148.34 
148.46 

•567  3543 
.568  2885 

I55-64 
155.76 

.624  7956 
.625  7776 

163.60 
163.74 

.685  2384 
.686  2728 

172-33 
172.48 

50 

•5'4  4375 

148.58 

.569  2235 

155.89 

.626  7604 

163.88 

.687  3082 

172.64 

51 

«5'5  3294 

148.70 

.570  1592 

156.02 

.627  7441 

164.02 

.688  3445 

172.79 

52 

.516  2219 

148.81 

•57'  0957 

156.15 

.628  7287 

164.16 

.689  3817 

172.94 

53 

•5'7  "5i 

148.93 

.572  0330 

156.27 

.629  7140 

164.30 

.690  4198 

173.10 

54 

.518  0090 

149.05 

.572  9710 

156.40 

.630  7002 

164.44 

.691  4588 

'73-25 

55 

.518  9037 

149.17 

•573  9098 

156.53 

.631  6873 

164.58 

.692  4988 

I7340 

56 

•5'9  799° 

149.28 

•574  8494 

156.66 

.632  6751 

164.72 

.693  5397 

173.56 

57 

58 
59 

.520  6951 
.521  5918 
.522  4893 

149.40 
149.52 
149.64 

•575  7897 
•576  73°8 
•577  6727 

156.79 
156.92 
'57-°4 

•633  6638 
•634  6534 
.635  6438 

164.86 
165.00 
165.14 

•694  5815 
.695  6243 
.696  6680 

'73-7' 
173.87 
174.02 

60 

•523  3875 

'49-75 

.578  6154 

I57-I7 

.636  6351 

165.28 

.697  7126 

174.18 

605 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

160° 

161° 

162° 

163° 

log  M. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0' 

3.697  7126 

174.18 

3.762  1519 

183.99 

3.830  3147 

194.87 

3.902  6107 

207.00 

1 

2 

.698  7581 
.699  8046 

'74-34 
174.49 

.763  2584 
.764  3639 

184.16 
I84-34 

.831  4845 
.832  6554 

195.06 
195.25 

.903  8534 
.905  0973 

207.21 
207.43 

3 

.700  8520 

174.65 

.765  4704 

184.51 

•833  8275 

'95-44 

.906  3425 

207.64 

4 

.701  9003 

174.80 

.766  5780 

184.68 

.835  0008 

195.64 

.907  5890 

207.86 

5 

3.702  9496 

174.96 

3.767  6867 

184.86 

3.836  1752 

195.83 

3.908  8368 

208.08  i 

6 

.703  9999 

175.12 

.768  7963 

185.03 

.837  3508 

196.02 

.910  0859 

208.29 

7 

.705  0511 

175.28 

.769  9070 

185.20 

.838  5275 

196.22 

.911  3363 

208.51 

8 

.706  1032 

'75-43 

.771  0187 

185.38 

•839  7°54 

196.41 

.912  5880 

208.72 

9 

.707  1562 

'75-59 

.772  1315 

I85.55 

.840  8844 

196.60 

.913  8410 

208.94 

10 

3.708  2102 

'75-75 

3-773  2454 

185.73 

3.842  0646 

196.80 

3-9'5  °953 

209.16 

11 
12 
13 

.709  2652 
.710  3211 

•7"  378o 

175-9' 
176.07 
176.22 

•774  36°3 
•775  4762 
.776  5932 

185.90 
186.08 
186.25 

.843  2460 
.844  4286 
.845  6133 

196.99 
197.19 
'97-38 

.916  3509 
.917  6078 
.918  8661 

209.38 
209.60 
209.81 

14 

.712  4358 

176.38 

•777  7"2 

186.43 

.846  7972 

197.58 

.920  1256 

210.03 

15 
16 

3.713  4946 
•7'4  5543 

176.54 
176.70 

3.778  8303 
•779  95°5 

186.60 
186.78 

3-847  9833 
.849  1705 

197.78 
197.97 

3.921  3865 
.922  6487 

210.25 
210.48 

17 

.715  6150 

176.86 

.781  0717 

186.96 

.850  3589 

198.17 

.923  9122 

210.70 

18 

.716  6766 

177.02 

.782  1940 

187.14 

.851  5486 

198-37 

.925  1770 

210.92 

19 

.717  7392 

177-18 

•783  3'74 

187.31 

.852  7394 

198.57 

.926  4432 

211.14 

20 
21 

3.718  8028 
.719  8673 

'77-34 
177.50 

3.784  4418 
.785  5672 

187.49 
187.67 

3-853  93'4 
.855  1245 

198.76 
198.96 

3.927  7107 
.928  9795 

211.36 
211.58 

22 

.720  9328 

177-66 

.786  6938 

187.85 

.856  3189 

199.16 

•93°  2497 

211.  8l 

23 

.721  9993 

177-83 

.787  8214 

188.03 

•857  5'45 

199.36 

.931  5212 

212.03 

24 

.723  0668 

178.00 

.788  9501 

188.21 

.858  7112 

199.56 

.932  7940 

212.25 

25 

3.724  1352 

178-15 

3.790  0799 

188.39 

3.859  9092 

199.76 

3.934  0682 

212.48 

26 
27 

.725  2045 
.726  2749 

178.31 
178.47 

.791  2108 
.792  3427 

188.57 
188.75 

.861  1084 
.862  3087 

199.96 

200.16 

•935  3438 
.936  6207 

212.70 
212.93 

28 
29 

.727  3462 
.728  4185 

178.63 
178.80 

•793  4757 
.794  6098 

188.93 
189.1! 

.863  5103 
.864  7131 

200.36 
200.56 

•937  8989 
•939  '785 

213.15 
213.38 

30 

3.729  4918 

178.96 

3-795  745° 

189.29 

3.865  9171 

200.77 

3.940  4595 

213.61 

31 

.730  5661 

179.13 

.796  8812 

189.47 

.867  1223 

200.97 

.941  7418 

213.83 

32 

.731  6413 

179.29 

.798  0186 

189.65 

.868  3287 

201.17 

•943  0254 

214.06 

33 

.732  7176 

'79-45 

•799  '57' 

189.83 

•869  5363 

201.37 

•944  3'°5 

214.29 

34 

•733  7948 

179.62 

.800  2966 

190.01 

.870  7452 

201.58 

•945  5969 

214.52 

35 

3.734  8730 

I79-78 

3.801  4372 

190.20 

3-871  955* 

201.78 

3.946  8847 

214.74 

36 

•735  95^2 

'79-95 

.802  5790 

190.38 

.873  1665 

201.98 

.948  1738 

214.97 

37 

.737  0324 

180.11 

.803  7218 

190.56 

•874  379' 

202.19 

.949  4644 

215.20 

38 

•738  H36 

180.28 

.804  8657 

190.65 

.875  5928 

202.39 

.950  7563 

215.43 

39 

•739  '957 

180.45 

.806  0108 

190.93 

.876  8078 

202.60 

.952  0496 

215.66 

40 

3.740  2789 

180.61 

3.807  1569 

191.11 

3.878  0240 

202.80 

3-953  3443 

216.90 

41 
42 

•74'  363' 
.742  4482 

180.78 
180.94 

.808  3041 
.809  4525 

191.30 
191.48 

.879  2414 
.880  4601 

203.01 
203.22 

•954  6403 
•955  9378 

216.13 
216.36 

43 

•743  5344 

i8i.ii 

.810  6020 

191.67 

.881  6800 

203.42 

.957  2366 

216.59 

44 

.744  6216 

181.28 

.811  7525 

191.86 

.882  9012 

203.63 

•958  5369 

216.82 

45 

3.745  7097 

181.45 

3.812  9042 

192.04 

3.884  1236 

203.84 

3-959  8385 

217.06 

46 

.746  7989 

181.61 

.814  0570 

192.23 

.885  3473 

204.05 

.961  1416 

217.29 

47 

•747  8891 

181.78 

.815  2110 

192.41 

.886  5722 

204.26 

.962  4460 

217.53 

48 

•748  9803 

181.95 

.8l6  3660 

192.60 

•887  7983 

204.46 

.963  7519 

217.76 

49 

.750  0725 

182.12 

.817  5222 

192.79 

.889  0257 

204.67 

.965  0592 

218.00 

50 

3.751  1657 

182.29 

3.818  6795 

192.98 

3.890  2544 

204.88 

3.966  3678 

218.23 

51 

.752  2599 

182.46 

.819  8379 

193.16 

.891  4843 

205.09 

.967  6779 

218.47  j 

52 

•753  355* 

182.63 

.820  9974 

'93-35 

.892  7155 

205.31 

.968  9895 

218.70 

53 

.754  4514 

182.80 

.822  1581 

193-54 

.893  9480 

205.52 

.970  3024 

218.94 

54 

•755  5487 

182.97 

.823  3199 

'93-73 

.895  1817 

205.73 

.971  6168 

219.18 

55 
56 
57 

3.756  6470 

•757  7464 
•758  8467 

183.14 
183.31 
183.48 

3.824  4829 
.825  6470 
.826  8122 

193.92 
194.11 
'94-3° 

3.896  4167 
.897  6529 
.898  8905 

205.94 
206.15 
206.36 

3.972  9326 
.974  2498 
•975  5684 

219.42 
219.66 
219.90 

58 
59 

•759  948i 
.761  0505 

183.65 
183.82 

.827  9785 
.829  1460 

194.49 
194.68 

.900  1293 
.901  3694 

206.57 
206.79 

.976  8885 

.978  2100 

220.13 
220.37 

60 

3.762  1539 

183.99 

3.830  3147 

194.87 

3.902  6107 

207.00 

3-979  533° 

220.61 

606 


TABLE  VI. 

For  findine  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

164° 

165° 

166° 

167° 

log  M. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  l". 

o 

1 

3-979  533° 
-980  8574 

220.62 

220.86 

4.061  6673 
.063  0842 

236.01 
236.28 

4.149  7198 
.151  2422 

253-57 
253.88 

4-244  5537 
.246  1975 

27A.I4 

2 
3 
4 

.982  1833 
.983  5106 
.984  8394 

221.10 
221-34 
221.58 

.064  5027 
.065  9229 
.067  3447 

236.56 
236.83 
237.11 

.152  7664 
.154  2925 
.155  8205 

254.19 
254.51 

•247  8434 
.249  4916 
.251  1419 

274.51 

5 
6 

7 
8 
9 

3.986  1696 
.987  5013 
•9*8  8345 
.990  1691 
.991  5051 

221.83 
222.07 
222.31 
222.56 
222.80 

4.068  7682 
.070  1933 
.071  6201 
.073  0486 
.074  4787 

237-39 
237.66 

237-94 
238.22 
238.50 

4-157  35°4 
.158  8822 
.160  4159 
.161  9515 
.163  4891 

255.14 

255.46 
255.78 
256.10 
256.42 

4.252  7944 
.254  4491 
.256  1061 
.257  7652 
.259  4266 

275.60 

275-97 
276.34 
276.71 

10 
11 

3.992  8427 
•994  1817 

223.05 
223.29 

4.075  9106 
.077  3441 

238.78 
239.06 

4.165  0285 
.166  5699 

256.74 
257.06 

4.261  0902 
.262  7560 

277-45 
277.82 

12 

.995  5222 

223.54 

.078  7792 

239-34 

.168  1132 

.264  4240 

'  L 

278.20 

13 
14 

.996  8642 
.998  2077 

223.79 
224.03 

.080  2161 
.081  6546 

239.62 
239.90 

.169  6585 
.171  2056 

257.70 
258.02 

.266  0943 
.267  7669 

278.95 

15 
16 
17 

3-999  5527 
I..OOQ  8991 
.002  2471 

224.28 

224-53 
224.78 

^.083  0948 
.084  5368 
.085  9804 

240.18 
240.46 
240.75 

4.172  7547 
.174  3058 
.175  8588 

258.35 
258.67 
259.00 

4.269  4417 
.271  1187 
.272  7981 

279.70 
280.08 

18 

.003  5965 

225.03 

.087  4257 

241.03 

.177  4138 

259  33 

.274  4797 

280.46 

19 

.004  9474 

225.28 

.088  8728 

241.32 

.178  9707 

259.65 

.276  1635 

280.84 

20 

4.006  2999 

22553 

..090  3215 

241.60 

^.180  5296 

259-98 

4.277  8497 

281.22 

21 

.007  6538 

225.78 

.091  7720 

241.89 

.182  0905 

260.31 

•279  5381 

281.60 

22 

.009  0093 

226.04 

.093  2242 

242.08 

.183  6534 

260.64 

.281  2289 

281.98 

23 
24 

.010  3663 
.on  7248 

226.29 
226.54 

.094  6781 
.096  1337 

242.56 
242.75 

.185  2182 
.186  7850 

260.97 
261.30 

.282  9219 
.284  6173 

282.36 

25 

4.013  0848 

226.79 

4.097  5911 

243.04 

4.188  3538 

261.63 

..286  3149 

283.14 

26 

.014  4463 

227.05 

.099  0502 

243  33 

.189  9246 

261.96 

.288  0149 

283.52 

27 

.015  8093 

227.30 

.100  5110 

243.62 

.191  4974 

262.30 

.289  7172 

283.91 

28 
29 

.017  1739 
.018  5400 

227.55 
227.81 

.101  9736 
.103  4379 

243-91 
244.20 

.193  0722 
.194  6490 

262.63 
262.97 

.291  4218 
.293  1288 

284.30 
284.69 

30 
31 

4.019  9077 
.021  2769 

228.06 
228.32 

.104  9040 
.106  3718 

244.49 

244.78 

4.196  2278 
.197  8086 

263.30 
263.64 

4.294.  838l 
.296  5498 

285.08 
285.47 

32 

.022  6476 

228.58 

.107  8414 

245.08 

•199  39*5 

263.98 

.298  2638 

285.87 

33 

.024  0199 

228.84 

.109  3127 

245-37 

.200  9764 

264.32 

.299  9802 

286.26 

34 

.025  3937 

229.09 

.no  7858 

245.67 

•202  5633 

264.66 

.301  6990 

286.66 

35 

.026  7691 

229.35 

.112  2607 

245.96 

4.204  1523 

265.00 

4.303  4201 

287.05  ' 

36 
37 

.028  1460 
.029  5245 

229.62 
229.88 

•JI3  7374 
.115  2158 

246.26 
246.55 

•205  7433 
.207  3363 

265-34 
265.68 

.305  1436 
.306  8695 

287.45 
287.85 

38 

.030  9045 

230.14 

.116  6960 

246.85 

.208  9314 

266.02 

.308  5978 

288.25 

39 

.032  2861 

230.40 

.118  1780 

247.15 

.2IO  5286 

266.37 

.310  3285 

288.65 

40 

.033  6693 

230.66 

.119  6618 

247.45 

.212  1278 

266.71 

.312  0616 

289.05 

41 

.035  0540 

230.92 

.121  1474 

247-75 

.213  7291 

267.06 

.313  7971 

289-45 

42 

.036  4404 

231.18 

.122  6348 

248.05 

.215  3325 

267.40 

•3'5  535° 

289.86 

43 

.037  8283 

231.45 

.124  1239 

248-35 

.216  9379 

267-75 

•3'7  2753 

290.26 

44 

.039  2177 

231.71 

.125  6149 

248.65 

.218  5455 

268.10 

.319  0181 

290.67 

45 

.040  6088 

231.97 

.127  1077 

248-95 

.220  1551 

268.44 

.320  7633 

291.07 

46 

.042  0015 

232.24 

.128  6023 

249.25 

.221  7668 

268.79 

.322  5110 

291.48  ; 

47 
1  48 

•°43  3957 
.044  7915 

232.51 
232.77 

.130  0988 
.131  5970 

249.56 
249.86 

.223  3806 
.224  9965 

269.14 
269.50 

.324  2611 
.326  0137 

291.89  I 
292.30 

49 

.046  1890 

233.04 

.133  0971 

250.17 

.226  6146 

269.85 

.327  7688 

292.71 

50 

.047  5880 

233-31 

.134  5990 

25047 

.228  2347 

270.20 

.329  5263 

293-13 

51 

.048  9887 

233-57 

.136  1028 

250.78 

.229  8570 

270.55 

.331  2863 

293-54 

52 

.050  3909 

233.84 

.137  6084 

251.08 

.231  4814 

270.91 

•333  °487 

293-95 

53 

.051  7948 

234.11 

.139  1158 

25I-39 

.233  I079 

271.27 

•334  8137 

294.37 

54 

053  2003 

234-38 

140  6251 

251.70 

.234  7366 

271.62 

•  336  5812 

294.79 

55 

054  6074 

234.65 

.142  1362 

252.01 

236  3674 

271.98 

.338  3511 

295.20 

56 

056  0161 

234.92 

143  6492 

252.32 

238  0003 

272.34 

.340  1236 

295.62 

57 

057  4264 

235-19 

145  1641 

252.63 

239  6354 

272.70 

.341  8986 

296.04 

58 

058  8384 

235-46 

146  6808 

252.94 

241  2727 

273.06 

.343  6762 

296.47 

59 

060  2520 

235.73 

I48  I994 

253-25 

242  9121 

273.42 

•345  4562 

296.89 

6O 

..061  6673 

236.01 

I49  7198    253.57 

244  5537 

273-78 

•347  *388 

297.31 

607 


TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

168° 

169° 

170° 

171° 

logM. 

Diff.  I". 

logM.    |  Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

O' 

4-347  *388 

297.31 

4.459  1242 

325.07 

4.581  9445 

358.31 

4.717  9835 

398.87  j 

1 

2 

.349  0240 
.350  8117 

*97-74 
298.16 

.461  0761 
.463  0311 

3*5-57 
326.08 

.584  0962 
.586  2516 

358.92 
359-53 

.720  3790 
.722  7790 

399.62  ; 
400.38 

3 

.352  6019 

298.59 

.464  9891 

326.59 

.588  4106 

360.15 

•7*5  1835 

401.14 

4 

•354  3948 

299.02 

.466  9501 

327.10 

•59°  5734 

360.76 

.727  5926 

401.90 

5 

4.356  1902 

299.45 

4.468  9142 

327.61 

4.592  7398 

361.38 

4.730  0063 

402.66 

6 

.357  9882 

299.88 

.470  8814 

328.12 

•594  910° 

362.00 

.732  4245 

403-43 

I   7 

.359  7888 

300.31 

.472  8517 

328.64 

•597  0838 

362.62 

•734  8474 

4°4  ^9  i 

8 

.361  5919 

300.75 

•474  8250 

3*9-i5 

•599  *6i5 

363-*5 

•737  *749 

404.96  . 

9 

.363  3977 

301.18 

.476  8015 

329.67 

.601  4428 

363.88 

•739  7°7° 

405.74 

1O 

4.365  2061 

301.62 

4.478  7811 

330.19 

4.603  6280 

364.50 

4.742  1438 

406.52 

11 

.367  0171 

302.05 

.480  7637 

330-7i 

.605  8169 

365-14 

•744  585* 

407.30 

12 

.368  8308 

302.49 

.482  7495 

33'-*3 

.608  0096 

365-77 

.747  0314 

408.08 

13 

.370  6470 

302.93 

•484  7385 

331-75 

.610  2061 

366.40 

•749  4822 

408.87 

14 

.372  4659 

303.37 

.486  7306 

332.28 

.612  4064 

367.04 

•751  9378 

409.66 

15 

4-374  2875 

303.81 

4.488  7258 

33*-8i 

4.614  6106 

367.68 

4-754  398i 

410.45 

16 

.376  1117 

304.26 

.490  7242 

333-33 

.616  8186 

368.32 

•756  8632 

411.24 

17 

•377  9386 

304.70 

.492  7258 

333.86 

.619  0304 

368.96 

•759  333° 

412.04 

18 

•379  768i 

3°5-!5 

•494  73°6 

334-4° 

.621  2461 

369.61 

.761  8077 

412.84 

19 

.381  6003 

3°5-59 

.496  7386 

334-93 

.623  4657 

370.26 

.764  2872 

4I3-65 

20 

4-383  435* 

306.04 

4.498  7498 

335-46 

4.625  6892 

370.91 

4.766  7715 

414.46 

21 

•385  *7*8 

306.49 

.500  7642 

336.00 

.627  9166 

37I-56 

.769  2606 

415.27 

22 

-387  "31 

306.94 

.502  7818 

336.54 

.630  1480 

372.21 

•771  7547 

416.08 

23 

.388  9561 

3°7-39 

.504  8026 

337.08 

.632  3832 

37*-87 

•774  *536 

416  90 

24 

.390  8019 

307.85 

.506  8267 

337.62 

.634  6224 

373-53 

•776  7574 

417.72 

25 

4.392  6503 

308.30 

4.508  8541 

338.16 

4.636  8656 

374-19 

4.779  2662 

418.54  I 

26 

•394  5OI5 

308.76 

.510  8847 

338.71 

.639  1127 

374-86 

.781  7799 

4^9-37 

27 

•396  3554 

309.21 

.512  9186 

339.26 

.641  3639 

375-5* 

.784  2986 

420.20 

28 

.398  2121 

309.67 

•5'4  9558 

339.80 

.643  6190 

376.19 

.786  8222 

421.03 

29 

.400  0715 

310.13 

.516  9962 

34°-35 

.645  8781 

376.86 

.789  3509 

421.86 

30 

4-401  9337 

310.59 

4.519  0400 

340.91 

4.648  1413 

377-53 

4.791  8846 

422.70 

31 

.403  7986 

311.06 

.521  0871 

341-46 

.650  4085 

378.21 

•794  4*33 

4*3-54 

32 

.405  6667 

311.52 

.523  1376 

342.02 

.652  6798 

378-89 

.796  9671 

424.39 

33 

.407  5368 

3H-99 

.525  1913 

34*-57 

.654  9552 

379-57 

.799  5160 

425.24 

34 

.409  4102 

311-45 

.527  2484 

343-J3 

.657  2346 

380.25 

.802  0700 

426.09 

35 

4.411  2863 

312.92 

4.529  3089 

343-69 

4.659  5182 

380.93 

4.804  6291 

426.95 

36 

.413  1652 

3'3-39 

.531  3728 

344.26 

.661  8059 

381.62 

•807  1934 

427.81 

37 

.415  0469 

313-86 

•533  44°° 

344.82 

.664  0977 

382.31 

.809  7628 

428.67 

38 

.416  9315 

3'4-33 

•535  5106 

345-39 

.666  3936 

383.00 

.812  3374 

4*9-53 

39 

.418  8189 

3H-8o 

•537  5846 

345-95 

.668  6937 

383.70 

.814  9172 

430.40 

40 

4.420  7091 

315-28 

4-539  6620 

346.52 

4.670  9980 

38439 

4.817  5022 

431.28 

41 

.422  6022 

3'5-75 

.541  7429 

347-09 

.673  3064 

385-09 

.820  0925 

432.15 

42 

.424  4982 

316.23 

•543  8272 

347-67 

.675  6191 

385.80 

.822  6881 

433.03 

43 

.426  3970 

316.71 

•545  9H9 

348.24 

•677  936° 

386.50 

.825  2889 

433-91 

44 

.428  2987 

317-19 

.548  0061 

348.82 

.680  2571 

387.21 

.827  8950 

434.80 

45 

4.430  2033 

317.67 

4.550  1007 

349.40 

4.682  5825 

387-9* 

4.830  5065 

435.69 

46 

.432  1108 

318.16 

.552  1989 

349.98 

.684  9121 

388.63 

•833  !*34 

436.59 

47 

.434  0212 

318.64 

•554  3°°5 

350-56 

.687  2460 

389-34 

•835  7456 

437.48 

48 

•435  9345 

3I9-I3 

.556  4056 

35i.i5 

.689  5842 

390.06 

•838  373* 

438-38 

49 

•437  8507 

319.61 

•558  5H3 

351-73 

.691  9268 

390.78 

.841  0062 

439.29 

50 

4-439  7698 

320.10 

4.560  6264 

35*-3* 

4.694  2736 

391.50 

4.843  6446 

440.20  : 

51 

.441  6919 

320.59 

.562  7421 

352.91 

.696  6248 

392.23 

.846  2886 

441.11 

52 

•443  6l69 

321.08 

.564  8614 

353-5° 

.698  9803 

392.96 

.848  9380 

442.03 

53 

•445  5449 

321.58 

.566  9842 

354-io 

.701  3402 

393-68 

.851  5929 

442.95 

54 

•447  4758 

322.07 

.569  1106 

354-69 

.703  7046 

394.42 

•854  *533 

443.87 

55 

4.449  4097 

322.57 

4.571  2405 

355-*9 

4.706  0733 

395-15 

4.856  9193 

444.80 

56 

.451  3466 

323.06 

•573  374i 

355-89 

.708  4464 

395.89 

.859  5909 

44IH 

57 

•453  *865 

323.56 

•575  5IJ3 

356.49 

.710  8240 

396-63 

.862  2680 

446.66 

58 

.455  2294 

324.06 

•577  6521 

357-io 

.713  2060 

3973« 

.864  9508 

447.60 

59 

•457  »753 

324.56 

•579  7965 

357.70 

•7i5  59*5 

398.12 

.867  6392 

448.54 

60 

4.459  1242 

3*5-07 

4.581  9445 

358.31 

4.717  9835 

398.87 

4.870  3333 

449-49 

TABLE  VI. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

172° 

173° 

174° 

175° 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0' 

4-870  3333 

449-49 

5.043  3285 

5H-47 

5.243  3165 

601.00 

5.480  1373 

722.00 

1 

.873  0331 

450.44 

.046  4191 

5I5-7I 

.246  9276 

602.69 

.484  4765 

7*4-4* 

2 

.875  7386 

451-39 

.049  5171 

516.96 

.250  5488 

604.38 

.488  8304 

726.87 

3 

.878  4499 

45*-35 

.052  6226 

518.21 

.254  1802 

606.08 

.493  1989 

7*9-33 

4 

.881  1668 

453-31 

•055  7356 

5I9-47 

.257  8218 

607.80 

•497  5823 

731.80 

5 
6 

4.883  8896 
.886  6182 

454.28 
455-*5 

5.058  8562 
.061  9843 

5*o-73 

522.00 

5.261  4738 
.265  1361 

609.53 
611.26 

5.501  9806 
.506  3939 

734-3° 
736.81 

7 

.889  3526 

456.23 

.065  1202 

523.28 

.268  8089 

613.00 

.510  8223 

739-33 

8 

.892  0929 

457.20 

.068  2637 

524.56 

.272  4922 

614.75 

.515  2659 

741-87 

9 

.894  8391 

458.19 

.071  4149 

5*5-85 

.276  i860 

616.52 

•519  7*4» 

744-44 

10 

4.897  5912 

459-17 

5.074  5738 

5*7-14 

5.279  8904 

618.29 

5.524  1992 

747.02 

11 

.900  3492 

460.16 

.077  7406 

528.44 

.283  6055 

620.08 

.528  6890 

749.61 

12 

.903  1132 

461.16 

.080  9151 

5*9-75 

.287  3313 

621.87 

•533  1946 

75*-*3 

13 

.905  8831 

462.16 

.084  0976 

531.06 

.291  0680 

623.67 

•537  7158 

754.86 

14 

.908  6591 

463.16 

.087  2879 

53*-38 

.294  8154 

625.49 

•54*  *5*9 

757-51 

15 

4.911  4411 

464.17 

5.090  4862 

533-71 

5.298  5738 

627.31 

5.546  8060 

760.18 

16 

.914  2291 

465.18 

.093  6924 

535-04 

.302  3432 

629.15 

•55i  375i 

762.87 

17 

.917  0233 

466.20 

.096  9067 

536-38 

.306  1237 

631.00 

•555  9605 

765.58 

18 

.919  8235 

467.22 

.100  1290 

537-73 

•3°9  915* 

632.85 

.560  5621 

768.31 

19 

.922  6299 

468.25 

•i°3  3594 

539.08 

•313  7179 

634.72 

.565  1802 

771.05 

20 
21 

4-9*5  44*5 
.928  2612 

469.28 

470.  3  i 

5.106  5980 
.109  8447 

540.44 
541.81 

5.317  5319 
.321  3571 

636.60 
638.49 

5.569  8148 
•574  4661 

773-8* 
776.61 

22 

.931  0862 

471-35 

.113  0997 

543-18 

•3*5  1938 

640.39 

•579  1341 

779-41 

23 

•933  9174 

472.39 

.116  3629 

544.56 

.329  0418 

642.30 

.587  8190 

782.24 

24 

.936  7549 

473-44 

.119  6344 

545-95 

•33*  9014 

644.23 

.588  5210 

785.08 

25 

4-939  5987 

474-49 

5.122  9143 

547-34 

5.336  7726 

646.16 

5.593  2401 

787-95 

26 

.942  4489 

475-55 

.126  2026 

548.74 

•340  6554 

648.11 

•597  9764 

790.84 

27 

•945  3°53 

476.61 

.129  4992 

550-15 

•344  5499 

650.07   .602  7302 

793-75 

28 

.948  1682 

477-68 

.132  8044 

551-57 

.348  4562 

652.04   .607  5014 

796.68 

29 

•95*  °375 

478.75 

.136  1181 

55*-99 

•35*  3744 

654.02 

.612  2903 

799.63 

30 

4-953  9i3* 

479-83 

5.139  4403 

554-4* 

5-356  3°45 

656.01 

5.617  0970 

802.60 

31 

•956  7954 

480.91 

.142  7711 

555-86 

.360  2466 

658.02 

.621  9216 

805.60 

32 

.959  6841 

481.99 

.146  1  1  06 

557-3° 

.364  2007 

660.04 

.626  7642 

808.62 

33 

.962  5793 

483.08 

.149  4588 

558.75 

.368  1671 

662.07 

.631  6250 

811.66 

34 

.965  4811 

484-18 

.152  8157 

5  6'o.  2  1 

.372  1456 

664.11 

.636  5041 

814.72 

35 
36 

4.968  3894 
.971  3044 

485.28 
486.38 

5.156  1813 
•159  5558 

561.68 
563.16 

5.376  1364 
.380  1396 

666.17 
668.24 

5.641  4017 
.646  3179 

817.81 
820.92 

37 

.974  2260 

487.49 

.162  9392 

564.64 

•384  1553 

670.32 

.651  2528 

824.05 

38 

•977  1543 

488.61 

.166  3315 

566.13 

.388  1834 

672.41 

.656  2065 

827.21 

39 

.980  0893 

489.73 

.169  7328 

567-63 

.392  2242 

674.52 

.661  1793 

830.39 

40 
41 

4.983  0311 
•985  9795 

490.85 
491.98 

5-173  *43i 
.176  5624 

569.13 
570.65 

5.396  2777 
.400  3439 

676.64 
678.77 

5.666  1713 
.671  1825 

833.60 
836.83 

42 

.988  9348 

493.12 

.179  9908 

572.17 

.404  4229 

680.92 

.676  2132 

840.08 

43 

.991  8970 

494.26 

.183  4284 

573-70 

.408  5149 

683.08 

.681  2635 

843.36 

44 

•994  8659 

495.40 

.186  8752 

575-*4 

.412  6199 

685.25 

.686  3336 

846.67 

45 

4.997  8418 

496.55 

5.190  3312 

576.78 

5.416  7379 

687.44 

5.691  4236 

850.00 

46 

5.000  8246 

497-71 

.193  7966 

578.34 

.420  8692 

689.64 

.696  5337 

853-36 

47 

.003  8143 

498.87 

.197  2713 

579.90 

•4*5  OI36 

691.85 

.701  6640 

856.75 

48 
19 

.006  8m 
.009  8148 

500.04 
501.21 

.200  7554 
.204  2489 

581.47 
583-05 

.429  1714 
•433  34*7 

694.08 
696.33 

.706  8147 
.711  9860 

860.16 
863.60 

50 

5.012  8256 

502.39 

5.207  7520 

584.64 

5-437  5*74 

698.59 

5-717  1779 

867.06 

51 

.015  8435 

503-57 

.211  2646 

586.23 

.441  7258 

700.86 

.722  3908 

870.56 

52 

.018  8685 

«*   J  J  'f 

504.76 

.214  7868 

587.84 

•445  9378 

703.15 

.727  6247 

874.08 

53 

.021  9006 

5°5-95 

.218  1l86 

589.45 

.450  1636 

705.45 

-73*  8798 

I77'6* 

54 

•°*4  9399 

.221  86O2 

591-07 

.454  4032 

707.77 

.738  1563 

881.21 

55 

5.027  9864 

508.36 

5.225  4116 

59*-7i 

5.458  6568 

710.10 

5.743  4544 

884.82 

56 

.031  0402 

509-57 

.228  9727 

59435 

.462  9244 

712.45 

•748  774* 

888.46 

57 

.034  1013 

510.79 

.232  5437 

596.00 

.467  2062 

714.81 

•754  "59 

892.13 

58 

.037  1697 

512.01 

.236  1247 

597.66 

.471  5022 

717.19 

.759  4798 

895-81 

59 

.040  2454 

513.24 

.239  7156 

599.32 

•475  81*5 

719.59 

.764  8659 

899.56 

60 

5.043  3285 

5I4-47 

5.243  3165 

601.00 

5.480  1373 

722.00 

5.770  2745 

903.31 

609 


TABLE  VI. 

I  or  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V. 

176° 

177° 

178° 

179° 

logM. 

Diff.  1". 

logM. 

Diff.  V. 

logM. 

Diff.  1". 

logM. 

Diff.  1". 

0' 

5.770  2745 

903.3 

6.144  6*89 

1205.3 

6.672  5724 

1808.  8 

7-575  464° 

3619 

1 

•775  7°58 

907.1 

.151  8807 

I2I2.0 

.683  4709 

1824.0 

•597  3596 

3680 

2 

•781  1599 

910.9 

•»59  1733 

1218.8 

.694  4613 

1839.5 

.619  6295 

3744 

3 

.786  6370 

914.8 

.166  5070 

1225.7 

•7°5  5454 

1855.3 

.642  2868 

3809 

4 

.792  1374 

918.7 

.173  8823 

1232.7 

.716  7248 

1871.3 

.665  3452 

3877 

5 

5.797  6612 

922.6 

6.181  2997 

1239.8 

6.728  ooio 

1887.5 

7.688  8192 

3948 

6 

.803  2086 

926.6 

.188  7597 

1246.9 

•739  3758 

1904.1 

.712  7239 

4021 

7 

.808  7798 

930.6 

.196  2628 

1254.1 

.750  8509 

1921.0 

•737  0756 

4097 

8 

.814  3751 

934.6 

.203  8095 

1261.4 

.762  4279 

1938.2 

.761  8913 

4176  i 

9 

.819  9946 

938.6 

.211  4002 

1268.8 

•774  I09° 

1955.6 

.787  1889 

4257 

10 

5.825  6386 

942.7 

6.219  °354 

1276.3 

6.785  8958 

1973-4 

7.812  9876 

4343 

11 

•831  3°73 

946.8 

.226  7158 

1283.8 

•797  79°4 

1991.5 

•839  3°75 

443i 

12 

.837  0008 

951.0 

.234  4419 

1291.5 

.809  7946 

2OIO.O 

.866  1702 

4524 

13 

.842  7195 

955-2 

.242  2142 

1299.2 

.821  9106 

2028.  g 

.893  5986 

4620 

14 

.848  4634 

959-5 

.250  0333 

1307.1 

.834  1404 

2048.0 

.921  6170 

4720 

15 

5.854  2329 

963.7 

6.257  8997 

1315.0 

6.846  4863 

2067.5 

7.950  2513 

4825 

16 

.860  0282 

968.0 

.265  8139 

1323.0 

.858  9503 

2087.3 

7.979  5292 

4935 

17 

.865  8495 

972.4 

.273  7766 

1331.1 

.871  5348 

2107.6 

8.009  4802 

5050 

18 

.871  6970 

976.8 

.281  7884 

J339-4 

.884  2422 

2128.3 

.040  1361 

5170 

19 

.877  5710 

981.2 

.289  8499 

1  347-7 

.897  0749 

2149.4 

.071  5309 

5296 

20 

5.883  4717 

985-7 

6.297  9617 

1356.2 

6.910  0353 

2170.9 

8.103  7°TI 

5428 

21 

.889  3993 

990.2 

.306  1244 

1364.7 

.923  1261 

2192.8 

.136  6857 

5568 

22 

.895  3542 

994.8 

•3J4  3387 

J373-3 

.936  3498 

2215.2 

.170  5274 

57H 

23 

.901  3365 

999-4 

.322  6052 

1382.1 

•949  7°93 

2238.0 

.205  2717 

5869 

24 

.907  3465 

1004.0 

•33°  9*47 

1391.0 

.963  2073 

2261.4 

.240  9679 

6032 

25 

5.913  3845 

1008.7 

6-339  2977 

1400.0 

6.976  8466 

2285.2 

8.277  6700 

6204 

26 

.919  4507 

1013.4 

.347  7249 

1409.1 

6.990  6304 

2309.6 

.315  4361 

6387 

27 

•925  5454 

1018.1 

.356  2072 

1418.3 

7.004  5616 

2334-3 

•354  3298 

6580 

28 

.931  6688 

1022.9 

.364  7451 

1427.6 

.018  6437 

2359-7 

•394  4205 

6786 

29 

•937  8213 

1027.8 

•373  3395 

H37-I 

.032  8796 

2385-7 

.435  7842 

7004 

30 

5.944  0030 

1032.7 

6.381  9910 

1446.7 

7.047  2729 

2412.2 

8.478  5044 

7238 

31 

.950  2144 

1037.6 

.390  7005 

1456.4 

.061  8271 

2439.4 

.522  6731 

7488 

32 

.956  4556 

1042.6 

•399  4687 

1466.2 

.076  5458 

2467.1 

.568  3920 

7755 

33 

.962  7269 

1047.7 

.408  2965 

1476.2 

.091  4329 

2495.4 

.615  7739 

8042 

34 

.969  0287 

1052.9 

.417  1846 

1486.4 

.106  4921 

2524.5 

.664  9442 

8352 

35 

5-975  3613 

1058.0 

6.426  1337 

1496.7 

7.121  7276 

2554.2 

8.716  0431 

8686 

36 

.981  7249 

1063.2 

•435  J449 

1507.0 

•137  1434 

2584.6 

.769  2286 

9048 

37 

.988  1198 

1068.4 

.444  2191 

1517.6 

.152  7440 

2615.8 

.824  6779 

944  i 

38 

5.994  5464 

1073.7 

•453  3569 

1528.3 

.168  5336 

2647.6 

.882  5925 

9870 

39 

6.001  0050 

1079.1 

.462  5594 

1539.2 

.184  5171 

2680.4 

.943  2018 

10340 

40 

6.007  4958 

1084.5 

6.471  8275 

1550.2 

7.200  6993 

2713-9 

9.006  7690 

10857 

41 

.014  0192 

1089.9 

.481  1620 

1561.3 

.217  0850 

2748.3 

.073  5974 

11429 

42 

.020  5756 

1095.4 

.490  5641 

1572.6 

.233  6796 

2783.5 

.144  0401 

12064 

43 

.027  1652 

IIOI.O 

.500  0346 

1584.1 

.250  4884 

2819.7 

.218  5102 

12773 

44 

.033  7885 

1106.7 

.509  5746 

1595.8 

.267  5170 

2856.8 

•297  4963 

13572 

45 

6.040  4457 

1112.4 

6.519  1850 

1607.7 

7.284  7712 

2894.8 

9.381  5820 

14476 

46 

.047  1372 

1118.1 

.528  8669 

1619.6 

.302  2571 

2934-1 

.471  4711 

15510 

47 

.053  8634 

1123.9 

.538  6216 

1631.8 

.319  9810 

2974.2 

.568  0247 

16704 

48 

.060  6246 

1129.8 

.548  4499 

1644.2 

•337  9494 

3015.6 

.672  3106 

18096 

49 

.067  4212 

"35-7 

•558  353° 

1656.8 

.356  1692 

3058.1 

.785  6758 

19741 

50 

6.074  2535 

1141.7 

6.568  3320 

1669  6 

7.374  6475 

3101.7 

9.909  8535 

21715 

51 

.081  1219 

1147.7 

.578  3881 

1682.4 

•393  39l8 

3146.8 

10.047  1256 

24127 

52 

.088  0269 

1153.8 

.588  5227 

1695.6 

.412  4099 

3193.0 

.200  5829 

27144 

53 

.oyj.  9687 

1  160.0 

.598  7368 

1708.9 

.431  7097 

3240.7 

•374  5584 

31023 

54 

.101  9479 

1166.3 

.609  0317 

1722.6 

.451  2999 

3289.9 

•575  3986 

36197 

55 

6.108  9647 

1172.6 

6.619  4°86 

1736.4 

7.471  1892 

334°-3 

10.812  9421 

4345° 

56 

.116  0196 

1179.0 

.629  8689 

1750.3 

.491  3870 

3392.6 

11.103  6719 

57 

.123  1131 

1185.4 

.640  4141 

1764.5 

.511  9029 

3446.5 

11.478  4880 

58 

.130  2455 

1192.0 

.651  0455 

1779.0 

.532  7472 

3502.1 

12.006  7617 

59 

•137  4173 

1198.6 

.661  7645 

1793-8 

•553  93°5 

3559-6 

12.909  8516 

60 

6.144  6289 

1205.3 

6.672  5724 

1808.  8 

7-575  4640 

3618.7 

610 


TABLE  VII. 

For  finding  the  True  Anomaly  in  a  Parabolic  Orbit  when  v  is  nearly  180°. 


w 

AO 

Diff. 

w 

AO 

Diff. 

w 

AO 

Diff. 

0              / 

/       // 

n 

0              f 

i         n 

II 

O             1 

/       // 

II 

155     0 

5 
10 
15 
20 
25 

3  23-°9 
19.74 
16.43 
13.17 

9-95 
6.77 

3-35 
3-3' 
3.26 
3.22 
3.18 
3.14 

160     0 

5 

10 
15 
20 
25 

I       6.70 

5-33 
3-97 
2.64 

i-33 

0.04 

:% 

•33 
•3' 

•2? 
26 

165     0 

10 
20 
30 
40 
50 

o  15.85 

14.98 

14.16 

13.38 

12.63 
11.91 

0.87 

0.82 
0.78 
0.75 
0.72 

0.69 

155   30 

35 
40 
45 
50 
55 

3     3-63 
0.54 

2  57-49 
54.48 
5i-5i 
48.58 

3.09 

3-°5 
3.01 
2.97 
2.93 
2.89 

160   30 

35 
40 
45 
50 
55 

o  58.78 
57-54 
56-31 
55-" 
53-93 
52.77 

.24 
•23 

.20 
.18 
.16 
.14 

166     0 

10 
20 
30 
40 
50 

0    11.22 

10.57 

9-95 
9.36 
8.80 
8.26 

0.65 
0.62 

0.59  1 

0.56    ; 
0.54    ' 
0.51 

156     0 

5 
10 
15 
20 
25 

2  45.69 
42.84 
40.03 
37.26 
34-53 
3i-83 

2.85 

2.8  1 

2.77 
2.73 
2.70 
2.66 

161     0 

5 
10 
15 
20 
25 

o  51.63 

50.50 
49.40 
48.32 
47.26 
46.21 

•13 
.10 
.08 
.06 

.05 
.02 

167     0 

10 
20 
30 

40 
50 

o     7.75 

l:ll 

6.37 
5.96 

5-57 

0.48 
0.46 
0.44 
0.41 

0-39 
0.37 

156  30 

35 
40 
45 
50 
55 

2    29.17 
26.55 

23-97 
21.43 
18.92 
16.44 

2.62 
2.58 
2.54 
2.51 
2.48 
2.44 

161    30 

35 
40 
45 
50 
55 

o  45.19 

44.18 

43-*9 
42.22 
41.26 
4°-33 

1.  01 

0.99 
0.97 

0.96 

o-93 
0.92 

168     0 

10 
20 
30 
40 
50 

o     5.20 
4.84 
4.51 
4.20 

3-9° 
3-62 

0.36 

°-33 
0.31 
0.30 
0.28 
0.26 

157     0 

5 

2    14.00 
I  I.'sO 

2.41 

162     0 

5 

o  39.41 

^8.ci 

o  90 

169     0 

10 

o     3-36 
3.11 

0.25 

10 
15 
20 
25 

11  Ox 
9.22 
6.89 
4.58 
2.31 

2.37 
2-33 
2.31 
2.27 
2.23 

10 
15 
20 
25 

J     3 

37.62 

36-75 

35-9° 
35.06 

0.89 
0.87 
0.85 
0.84 
0.82 

20 
30 
40 
50 

2.88 
2.66 
2.46 
2.27 

0.23 

0.22 
0.20 
0.19 

0.18 

157    30 

35 
40 
45 
50 
55 

2       0.08 

i  57-89 
55-71 
53-57 
51.46 

49-39 

2.19 
2.17 
2.15 

2.  II 

2.07 
2.04 

162   30 

35 
40 
45 
50 
55 

o  34.24 

33-43 
32.64 
31.86 
31.10 
3°-35 

0.8  1 
0.79 
0.78 
0.76 
0.75 
0.73 

170     0 

10 
20 
30 
40 
50 

o     2.09 

1.92 
1.76 

1.62 

1.48 
1.35 

0.17 
0.16 
0.14 
0.14 
0.13 

O.IZ 

158     0 

5 
10 
15 
20 
25 

i  47-35 
45-34 
43-35 
41-39 
39-47 
37-57 

2.01 
1-99 
1.96 
1.92 

I.gO 
1.87 

163     0 

5 
10 
15 
20 
25 

o  29.62 
28.90 
28.20 
27.51 
26.83 
26.16 

0.72 
0.70 
0.69 
0.68 
0.67 
0.65 

171     0 

10 
20 
30 
40 
50 

o     1.23 

1.  12 

1.02 
0.93 
0.84 
0.76 

O.I  I 
O.IO 
0.09 
0.09 
0.08 
0.08 

158   30 

35 
40 
45 
50 
55 

i   35«70 
33.87 
32.06 
30.28 
28.52 
26.80 

/ 

1.83 

1.81 

1.78 
1.76 
1.72 
1.70 

163   30 

35 
40 
45 
50 
55 

o  25.51 
24.88 
24.25 
23.64 
23.04 
22.45 

0.63 
0.63 
0.61 
0.60 
0.59 
0.57 

172     0 

10 
20' 
30 
40 
50 

o     0.68 
0.6  1 
0.55 
0.49 
0.44 
0.39 

0.07 
0.06 
0.06 
0.05 
0.05 
0.04 

159     0 

5 
10 
15 
20 
25 

i  25.10 

23-43 
21.78 
20.16 

18.57 
17.00 

1.67 
1.65 
1.62 
1.59 

i-57 

I.CC 

164     0 

5 
10 
15 
20 
25 

o  21.88 
21.31 
20.76 

20.22 
I9.69 
I9.l8 

0.57 
0.55 
0.54 

°-53 
0.51 
0.51 

173     0 

10 
20 
30 
40 
50 

o     0.35 
0.31 

0.27 
0.24 

0.21 
0.19 

0.04 
0.04 
0.03 
0.03 

O.O2 
0.03 

159  30 

35 
40 
45 
50 
55 

i   15.45 
13.94 
12.44 
10.97 

9-53 
8.10 

J  J 

1.51 

1.50 
1.47 
1.44 
1.43 

1.40 

164   30 

35 
40 
45 
50 
55 

o  18.6-7 

18.17 
17.69 
17.21 
16.75 

16.29 

0.50 
0.48 
0.48 
0.46 
0.46 
0.44 

174     0 
175     0 
176     0 
177     0 

178     0 
179     0 

o     o.i  6 
0.07 

O.O2 
O.OI 
0.00 
0.00 

0.09 
0.05 
O.OI 
O.OI 
O.OO 

o.oc 

160     0 

i     6.70 

165     0 

o  15-85 

180     0 

0      O.OO 

fill 


TABLE  Vfll. 

For  finding  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V 

log  N 

Diff. 

V 

log  J\" 

Diff. 

V 

log^T 

Diff. 

0    / 

0    / 

O    / 

0  0 

30 
1  0 
30 
2  0 

30 

0.025  5763 
.025  5749 
.025  5707 
.025  5638 
.025  5542 
.025  5418 

J? 

96 
I24 
152 

30  0 

30 
31  0 
30 
32  0 

30 

0.020  7913 
.O2O  6368 
.020  4802 
.O2O  3215 
.020  1607 

.019  9979 

'545 
1566 
1587 
1608 
1628 
1649 

60  0 

30 
61  0 
30 
62  0 

30 

0.008  8644 
.008  6458 
.008  4277 
.008  2103 
.007  9934 
.007  7774 

2186 

2181 

2174 

2169 
2160  I 

2153 

3  0 

,     30 
4  0 
30 
5  0 

30 

0.025  5*66 
.025  5087 
.025  4881 
.025  4647 
.025  4386 
.025  4097 

179 

2OO 

*34 
261 
289 
316 

33  0 

30 
34  0 
30 
35  0 

30 

0.019  8330 
.019  6662 
.019  4974 
.019  3267 
.019  1540 
.018  9795 

1668 
1688 
1707 
1727 

1745 
1765 

63  0 

30 
64  0 
30 
65  0 

30 

0.007  5621 
.007  3477 
.007  1343 
.006  9220 
.006  7108 
.006  5008 

2144 

2134 
2123 

2112 

2IOO 

2086 

6  0 

30 
7  0 
30 
8  0 

0.025  378  1 

•025  3437 
.025  3066 
.025  2668 
.025  2243 

344 
371 
398 
425 

36  0 

30 
37  0 
30 
38  0 

0.018  8030 
.018  6248 
.018  4448 
.018  2629 
.018  0794 

1782 
1800 
1819 
JJJ35 

66  0 

30 
67  0 
30 
68  0 

0.006  2922 
.006  0849 
.005  8792 
.005  6750 
.005  4725 

2073 

2057 

2042 
2025 

2008 

30 

.025  1791 

452 

30 

.017  8941 

1869 

30 

.005  2717 

1988 

9  0 

30 

0.025  i5  1  1 
.025  0805 

506 

39  0 

30 

0.017  7072 
.017  5186 

1886 

69  0 

30 

0.005  07*9 
.004  8760 

1969 

10  0 

.025  0271 

534 

40  0 

.017  3283 

1903 

70  0 

.004  68  i  i 

1949 

30 

.024  9711 

560 

30 

.017  1365 

1918 

30 

.004  4884 

1927 

11  0 

.024  9124 

1*7 

41  0 

.016  9432 

'933 

71  0 

.004  2980 

1904 

-  O  O  _ 

30 

.024  8510 

614 
641 

30 

.016  7483 

1949 
1963 

30 

.004  1  1  oo 

I  ooO 

1855 

12  0 

30 
13  0 
80 
14  0 

30 

0.024  7869 
.024  7201 
.024  6507 
.024  5786 
.024  5039 
.024  4266 

668 
694 

721 
747 
773 
800 

42  0 

30 
43  0 
30 
44  0 

30 

0.016  5520 
.016  3542 
.016  1550 
.015  9545 
.015  7526 
.015  5495 

1978 
1992 
2005 
2019 
2031 
2045 

72  0 
30 
73  0 

30 
74  0 
30 

0.003.  9*45 
.003  7416 
.003  5613 
.003  3839 
.003  2094 
.003  0380 

1829 
1803 

1774 
1745 
1714 
1682 

15  0 

30 
16  0 
30 
17  0 

30 

0.024  3466 
.024  2641 
.024  1789 
.024  091  1 
.024  0008 
.023  9079 

825 
852 
878 

9°  3 
929 

954 

45  0 

30 
46  0 
30 
47  0 

30 

0.015  3450 
.015  1394 
.014  9326 
.014  7247 
.014  5157 
.014  3057 

2056 
2068 
2079 
2090 

2100 
2110 

75  0 

30 
76  0 

30 
77  0 
30 

O.002  8698 
.002  7049 

.002  5433 

.002  3854 
.002  2311 

.002  0806 

1649 
1616 

1543 

1505 
1465 

18  0 

30 
19  0 

30 
20  0 

30 

0.023  8125 
.023  7145 
.023  6140 
.023  5109 
.023  4054 
.023  2973 

980 

1005 
1031  ! 

I055 
1081  I 
1105 

48  0 
30 
49  0 
30 
50  0 
30 

0.014  0947 
.013  8827 
.013  6698 
.013  4561 
.013  2416 
.013  0263 

2120 
2129 
2137 
2145 
2153 
2IDO 

78  0 
30 
79  0 

30 
80  0 
30 

o.ooi  9341 
.001  7917 
.001  6535 
.001  5196 
.001  3903 
.001  2656 

1424 
1382 

1339 
1293 
1247 
1198 

21  0 

30 
22  0 

0.023  1868 
.023  0738 
.022  9584 

1130 
1154 

51  0 

30 
52  0 

0.012  8103 
.012  5936 
,OI2  7764 

2167 
2I72 

81  0 

30 
82  0 

o.ooi  1458 
.001  0309 
.000  9211 

1149 

1098 

30 
23  0 

30 

.022  8405 
.022  72O2 

.022  5975 

1179 

1203 
1227 

30 
53  0 

30 

J  /   T 

.012  1585 

.on  9402 
.on  7215 

2179 
2183 
2187 

30 
83  0 

30 

.000  8l66 

.000  7175 
.000  6240 

1045 
991 

935 

x«vn 

1251 

2191 

oy  D 

24  0 

30 
25  0 

O.O22  4724 

.022  3449 

.022  2151 

1298 

54  0 

30 
55  0 

o.on  5024 
.on  2829 
.on  0632 

2195 

2197 

84  0 
30 
85  0 

o.ooo  5364 
.000  4546 
.000  3790 

818 

756 

30 

.022  0829 

1322 

30 

.010  8432 

22OO 

30 

.000  3096 

h  « 

26  0 

.021  9484 

I3tl 

56  0 

.010  6231 

22OI 

86  0 

.000  2468 

O2S 

30 

.021  8ll6 

1368 

30 

.010  4029 

22O2 

30 

.000  1906 

562 

1390 

2202 

493 

27  0 

0.021  6726 

57  0 

o.oio  1827 

87  0 

o.ooo  1413 

30 
28  0 
30 
29  0 

30 

.O2I  5312 
.021  3876 
.021  2418 
.021  0938 
.020  9436 

1414 

I436 
1458 
1480 
1502 

30 
58  0 
30 
59  0 

30 

.009  9625 
.009  7424 
.009  5225 
.009  3028 
.009  0834 

22O2 
2201 
2I99 
2197 
2I94 
2190 

30 
88  0 
30 
89  0 

30 

.000  0990 
.000  0639 
.000  0363 
.000  0163 
.000  0041 

423 

276 

200 
122 

41 

30  0 

0.020  7913 

60  0 

0.008  8644 

90  0 

O.OOO  0000 

612 


TABLE  VIII. 

For  finding  the  Time  from  the  Perihelion  in  a  Parabolic  Orbit. 


V 

log  N' 

Difif. 

V 

log  N' 

Diflf. 

V 

log  N' 

Diff. 

O    1 

0    1 

o   t 

90  0 
30 
91  0 

0.000  0000 

9.999  9876 

•999  95°7 

124 
369 

120  0 

30 
121  0 

9.963  1069 
.962  0074 
,960  8971 

10995 
11103 

150  0 

30 
151  0 

9.889  0321 
.887  8738 
.886  7259 

11583 
11479 

30 
92  0 

30 

•999  8893 
•999  8°39 
•999  6944 

614 

854 
1095 

J33X 

30 
122  0 

30 

•959  7764 
.958  6454 
.957  5046 

1  1  207 
11310 
11408 
11504 

30 
152  0 

30 

.885  5887 
.884  4627 
.883  3481 

11372 
11260  . 
11146 
11026 

93  0 

30 
94  0 
30 
1  95  0 

30 

9-999  5613 
•999  4°46 
.999  2246 
.999  0215 

•998  7955 
.998  5468 

1567 
1800 
2031 
2260 
2487 
2711 

123  0 
30 
124  0 
30 
125  0 
30 

9.956  3542 
•955  '945 
•954  0*58 
.952  8483 
.951  6624 
.950  4684 

11597 
11687 
11775 
11859 
11940 
12018 

153  0 

30 
154  0 
30 
155  0 

30 

9.882  2455 
.881  1552 
.880  0775 
.879  0129 
.877  9616 
.876  9242 

10903 
10777 
10646 
10513 
10374 
10232 

96  0 

30 
97  0 

30 
98  0 

30 

9.998  2757 
.997  9824 
.997  6669 

•997  3*97 
.996  9708 
.996  5906 

2933 
3155 
337^ 

3589 
3802 
4015 

126  0 

30 
127  0 
30 
128  0 

30 

9.949  2666 
•948  0573 
.946  8408 
•945  6174 
•944  3875 
•943  15*3 

12093 
12165 
12234 
12299 
12362 
12421 

156  0 

30 
157  0 
30 
158  0 

30 

9.875  9010 
.874  8922 
•873  8984 
.872  9198 
.871  9569 
.871  0099 

10088 
9938 
9786 
9629 
9470 
9307 

99  0 

30 
100  0 
30 
101  0 

30 

9.996  1891 
.995  7666 
•995  3*34 
•994  »596 
•994  3755 
•993  8712 

4225 

J$ 

4841 

5°43 
5242 

129  0 

30 
130  0 
30 
131  0 

30 

9.941  9092 
.940  6615 

•939  4°85 
.938  1506 
.936  8881 
•935  6213 

12477 
12530 
12579 
12625 
12668 
12707 

159  0 

30 
160  0 
30 
161  0 

30 

9.870  0792 
.869  1652 
.868  2683 
.867  3886 
.866  5266 
.865  6827 

9140 
8969 

8797 
8620 
8439 
8257 

103  0 

30 
103  0 
30 
104  0 

30 

9-993  347° 
.992  8031 
.992  2397 
.991  6570 
.991  0553 
•99°  4347 

5439 
5634 
5827 
6017 
6206 
6391 

132  0 

30 
133  0 
30 
134  0 

30 

9-934  35°6 
•933  0763 
.931  7987 
.930  5183 
.929  2353 
.927  9501 

12743 
12776 
12804 
12830 
12852 
12871 

162  0 
30 
163  0 
30 
164  0 
30 

9.864  8570 
.864  0500 
.863  2620 
.862  4932 
.861  7439 
.861  0145 

8070 
7880 
7688 

7493 
7294 
7092 

105  0 

30 
106  0 
30 
107  0 

30 

9.989  7956 
.989  1380 
.988  4622 
.987  7685 

•987  0571 
.986  3281 

6576 
6758 

6937 

7114 
7290 
7462 

135  0 

30 
136  0 
30 
137  0 

30 

9.926  6630 

•925  3745 
.924  0848 
.922  7943 
.921  5035 
.920  2126 

12885 
12897 
12905 
12908 
12909 
12906 

165  0 

30 
166  0 
30 
167  0 

30 

9.860  3053 
.859  6164 
.858  9482 
.858  3010 
.857  6750 
.857  0704 

6889 
6682 
j>47» 

O20O 
6046 
5829 

108  0 
30 
109  0 
30 
110  0 
30 

9.985  5819 
.984  8186 
.984  0385 
.983  2418 
.982  4288 
.981  5996 

7633 
7801 
7967 
8130 
8292 
84.0 

138  0 

30 
139  0 
30 
140  0 

30 

9.918  9220 
.917  6321 
.916  3433 
.915  0559 
.913  7703 
.912  4870 

12899 
12888 
12874 
12856 
12833 
12808 

168  0 

30 
169  0 
30 
170  0 

30 

9.856  4875 
.855  9266 
•855  3878 
•854  8714 
•854  3775 
.853  9065 

£! 

5164 

4939 
4710 
4481 

111  0 

30 
112  0 
30 
113  0 

30 

9.980  7545 
•979  8938 
•979  0177 
.978  1264 

.977  2202 
.976  2993 

TJ 

8607 
8761 
8913 
9062 
9209 
Q"K1 

141  0 

30 
142  0 
30 
143  0 

30 

9.911  2062 
.909  9283 
.908  6538 
.907  3831 
.906  1164 
.904  8542 

12779 
12745 
12707 
12667 
12622 
12573 

171  0 

30 
172  0 
30 
173  0 

30 

9.853  4584 
•853  °335 
.852  6319 
.852  2538 
.851  8994 
.851  5687 

4249 
4016 
378i 

3544 
3307  ' 
3067 

114  0 

30 
115  0 
30 
116  0 

30 

9-975  364° 
•974  4H5 
•973  45io 
•972  4739 
.971  4833 
.970  4796 

7  J  J  3 

9495 
9635 
9771 
9906 

10037 
10167 

144  0 

30 
145  0 
30 
146  0 

30 

9.903  5969 
.902  3449 
.901  0985 
.899  8582 
.898  6243 
•897  3972 

12520 
12464 
12403 
12339 
12271 
12198 

174  0 

30 
175  0 
30 
176  0 

30 

9.851  2620 
.850  9794 
.850  7209 
.850  4868 
.850  2770 
.850  0917 

2826 
2585 
2341 
2098 
1851  - 
1608  j 

117  0 

30 
118  0 
30 
119  0 

30 

9.969  4629 
.968  4337 
.967  3920 
.966  3382 
.965  2726 
.964  1954 

10292 
10417 
'°538 
10656 
10772 
10885 

147  0 

30 
148  0 
30 
149  0 

30 

9.896  1774 
.894  9652 
.893  7610 
.892  5652 
.891  3782 
.890  2004 

I2I22 

J2042 
II958 
II870 
JI778 

n6?3 

177  0 

30 
178  0 
30 
179  0 

30 

9.849  9309 
.849  7948 
.849  6833 
.849  5966 
.849  5346 
.849  4974 

1361 
1115  ; 
867 
620 

37*  ' 
124 

120  0 

9.963  1069 

150  0 

9.889  0321 

180  0 

9.849  4850 

613 


TABLE  IX. 

F>r  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  Orbits  of  great  eccentricity 


X 

A 

Difif. 

B 

Difif. 

c 

B' 

Difif. 

C' 

0 

n 

n 

it 

If 

II 

II 

It 

„ 

0 

O.OO 

0.000 

0.000 

o.ooo 

o.ooo 

1 

O.OO 

O.OO 

0.000 

o.ooo 

0.000 

0.000 

2 

0.0  1 

O.OI 

0.000 

0.000 

0.000 

o.oco 

3 

0.05 

0.04 

o.ooo 

o.ooo 

o.ooo 

o.ooo 

4 

0.12 

0.07 

0.000 

o.ooo 

0.000 

o.ooo 

O.I  I 

5 

0.23 

o.  1  6 

o.ooo 

0.000 

0.000 

o.ooo 

6 

0.35 

0.000 

o.ooo 

o.ooo 

0.000 

7 

0.02 

0.23 

o.ooo 

0.000 

0.000 

o.ooo 

8 

o-93 

0.31 

0.000 

o.ooo 

o.ooo 

o.ooo 

1        9 

1.77 

0.40 

o.ooo 

0.000 

0.000 

o.ooo 

J  J 

0.49 

1      10 

1.82 

o.ooo 

0.000 

0.000 

o.ooo 

11 

2.42 

o.6c 

0.000 

0.000 

0.000 

0.000 

12 

7.14 

0.72 

o.ooo 

0.000 

o.ooo 

o.ooo 

13 

J       T^ 

3-99 

0.85 

0.000 

0.000 

o.ooo 

0.000 

14 

4-99 

.00 
.14 

O.OO  I 

0.000 

0.001 

o.ooo 

15 

6.13 

0.001 

o.ooo 

O.OOI 

o.ooo 

16 

7.4-1 

.30 

O.OO2 

.001 

0.000 

0.00  1 

.000 

o.coo 

17 

/TO 
8.90 

f 

0.002 

.000 

o.ooo 

O.OO2 

.001 

o.ooo 

18 
19 

10.55 
12.40 

65 
.85 
2.05 

0.003 
0.004 

.001 
.001 
.001 

0.000 

o.ooo 

0.002 

0.003 

.000 

.001 

.001 

0.000 

o.oco 

20 
21 

14.45 
l6.7O 

2.25 

0.005 
0.006 

.001 

0.000 

o.ooo 

0.004 
0.005 

.001 

0.000 

o.ooo 

22 

I9.l8 

2.48 

0.008 

.002 

0.000 

0.006 

.001 

0.000 

23 

21.89 

2.71 

O.OIO 

.002 

o.ooo 

0.008 

.002 

o.ooo 

24 

X 

24.81 

2.94 

O.OI  2 

.002 

0.000 

O.OIO 

.002 

0.000 

T       J 

3.20 

.002 

.002 

25 

28.07 

0.014 

0.000 

0.012 

o.ooo 

26 

27 

m**        J 

31.48 

35.20 

3-45 
3-72 

T. 

0.017 

0.020 

.003 
.003 

0.000 
0.000 

0.014 
0.017 

.002 

.003 

o.ooo 

0.000 

28 
29 

39-'9 

43-47 

3-99 
428 

4-57 

0.025 
0.030 

.005 
.005 

.005 

0.000 
0.000 

O.O2O 

0.024 

.003 
.004 

004 

o.ooo 

0.000 

30 
31 
32 
33 
34 

48.04 
52.91 
58.09 

63-59 
69.42 

4.87 
5.18 

5-5° 
5.83 
6.15 

0.035 

0.041 

0.047 
0.055 

0.064 

.006 

.006 

.008 
.009 
.009 

0.000 
0.000 
0.000 
0.000 

o.ooo 

0.028 

0.033 
0.039 
0.045 

0.052 

.005 
.006 
.006 
.007 
.008 

o.ooo 

0.000 

o.ooo 

0.000 

0,000 

35 
36 
37 

38 

75-57 
82.07 
88.92 
96.12 

6.50 
6.85 
7.20 

0.073 
0.084 
0.096 
0.109 

.01  1 

.012 

.013 

0.000 

o.ooo 

0.000 

o.ooo 

0.060 
0.068 
0.078 
0.088 

.008 

.010 

.010 

o.ooo 

0.000 

o.ooo 

0.000 

39 

103.68 

7.56 
7-93 

0.123 

.014 

.016 

0.000 

O.1OO 

.012 

.013 

o.ooo 

40 
41 
43 
43 
44 

in.  61 

119.92 
128.62 
137.70 
147.18 

8.31 
8.70 
9.08 
9.48 

0.139 

0.156 

0.175 

0.196 
0.218 

.017 

.019 

.021 
.022 

0.000 

o.ooo 
o.ooo 
o.ooo 
o.ooo 

0.113 
0.127 
0.142 

0.159 
0.177 

.014 
.015 
.017 
.018 

0.000 

o.ooo 

0.000 
0.000 

o.ooo 

9.87 

.025 

.020 

45 
46 

47 
48 
49 

157.05 
167.34 
178.04 
189.16 

200.71 

10.29 
10.70 

II.  12 
11-55 

11.98 

0.243 

0.269 

0.298 
0.328 
0.361 

.026 
.029 
.030 

.033 

.036 

o.ooo 

0.000 

o.ooo 

0.000 

o.ooo 

0.197 

0.219 
0.242 
0.267 

0.294 

.022 

.023 
.025 
.027 

.029 

0.000 

o.ooo 

0.000 

o.ooo 
o.ooo 

5O 
51 
52 
53 
54 

212.69 
225  ic 

237-95 
251.25 
265.01 

12.41 

12.85 
13.30 
13.76 
14.20 

Q-397 

0.436 
0.477 

0.521 

0.567 

•°39 

.041 
.044 

.046 
.050 

o.ooo 

0.000 

O.OOJ 
0.00  1 
O.OO  I 

0.323 

0-354 

0.388 
0.424 

0.462 

.031 
.034 

.036 

.038 

.040 

0.000 

o.ooo 
o.ooo 

0.000 

o.ooo     j 

55 
56 
57 

58 
59 

279.21 
293.88 
309.02 
324.62 
340.70 

14.67 
I5.I4 

15.60 

16.08 
16.56 

0.617 
0.671 

0.727 

0.787 

0.851 

.054 

.056 

.060 
.064 

.068 

O.OO  I 
0.002 
O.OO2 
0.002 
O.OO2 

0.502 

0.546 
0.592 

0.641 

0.693 

.044 

.040 

.049 

.052 

.056 

o  ooo 

0001 
O  OOI 
C  OOI 

O.OOI 

60 

357-^6 

0.919 

0.003 

0.749 

O.OO2 

614 


TABLE  IX. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  Orbits  of  great  eccentricity. 


X 

A 

Diff. 

B 

Diff. 

c 

B1 

Diff. 

C' 

0 

n 

,/ 

„ 

n 

II 

n 

// 

n 

60 
61 
62 

357-26 

374-3° 
391.84 

17.04 

17-54 
18.02 

0.919 
0.990 
.066 

.071 
.076 

0.003 
0.003 
0.003 

0.749 
0.807 
0.869 

.058 
.062 
.066 

0.002 
0.002 
O.OO2 

63 
64 

409.86 
428.38 

18.52 

19.02 

•'45 
-230 

SI 
.088 

0.004 
0.004 

0.935 

1.004 

.069 
•°73 

0.002 
O.OO2 

65 

66 
67 
68 
69 

447.40 
466.92 
486.96 
507.51 
528.58 

19.52 

20.04 

2°-55 
21.07 
21.59 

.318 
.411 
.510 
.613 
.721 

.093 

.099 
.103 
.108 
.114 

0.004 
0.005 
0.005 
0.006 
0.006 

.077 
.154 

•235 
.321 
.411 

.077 
.081 
.086 
.090 
.094 

0.003 

o  003 
0.003 
0.004 
0.004 

70 
71 
72 
73 
74 

550.17 
572.29 

59494 
618.12 
641.85 

22.12 

22.65 
23.18 
23.73 
24.28 

.835 
•954 

2.078 
2.209 
2-345 

.119 
.124 

'.\\6 
•H3 

0.007 
0.007 
0.008 
0.009 
0.009 

.505 
.605 
.709 
.819 
•934 

.100 

.104 

.110 

.115 

.121 

0.004 
0.005 
0.005 
0.006 
0.006 

75 

666.13 

_          0 

2.488 

0.0  1  0 

2.055 

,-f. 

0.007 

76 

690.96 

24.83 

,  f    «O 

2.637 

.149 

O.OII 

2.181 

.1  2O 

0.007 

77 
78 
79 

716.34 
742.29 
768.81 

25.38 
25.95 
26.52 
27.09 

2-793 
2.956 
3.125 

.163 
.169 

.177 

0.012 

0.013 
0.014 

2.314 
2-453 
2-599 

•139 
.146 

•153 

0.008 

0.008 

0.009 

80 
81 
82 
83 
84 

795-9° 

85i'.84 
880.70 
910.16 

27.67 
28.27 
28.86 
29.46 
30.07 

3.302 
3.486 
3.678 
3.878 
4.087 

.184 
.192 

.200 
.209 
.216 

0.015 
0.016 
0.017 
0.018 

0.020 

2.752 
2.912 
3-°79 
3-255 
3-439 

.l6o 

S3 

.184 
.192 

0.010 
O.OII 
O.OI2 
0.013 
0.014 

85 
86 
87 
88 
89 

940.23 
970.92 
1002.24 
1034.20 
1066.  81 

30.69 

31.96 
32.6l 
33-27 

4-3°3 
4.529 
4.764 
5.008 
5.262 

.226 
•235 
•244 
.254 
.265 

0.021 
0.023 
0.024 
O.O26 
0.028 

3.631 

3-833 
4.044 
4.266 
4.498 

.Z02 
.211 
.222 
.232 
.243 

0.015 

0.016 
0.018 
0.019 

0.021 

90 

1100.08 

5  527 

0.030 

4.741 

ire1 

0.023 

91 
92 
93 
94 

1134.02 
1168.64 
1203.95 
~    1239.97 

33-94 
34.62 

35-31 
36.02 

36-75 

5.801 
6.087 
6.385 
6.694 

.298 

•3°9 
.322 

0.032 
0.034 
0.036 
0.038 

4.996 
5.263 

5-544 
5.838 

•255 
.267 
.281 
.294 
•3°9 

O.O25 
0.027 
0.029 
0.032 

95 
98 
97 

1276.72 
1314.21 

37-49 
38.24 

7.016 
7.350 
7.698 

•334 
.348 

0.041 
0.044 
0.047 

6.147 
6.471 
6.812 

.324 
•341 

"9  C"  A 

0.035 
0.038 

0.041 

98 

1391.46 

39.01 

8.060 

.362 

0.050 

7.171 

•359 
178 

0.045 

99 

1431.27 

40.61 

8.437 

•377 
•392 

0.053 

7-549 

•3/° 
•397 

0.049 

100    0 

30 
101     0 
30 
102    0 

30 

1471.88 
1492.50 

I534-38 
1555.64 

1577.12 

20.62 
20.83 
21.05 
21.26 
21.48 
21.70 

8.829 
9.032 
9.238 

9-449 
9.664 

9-883 

S3 

.211 

.215 
.219 
.225 

0.056 
0.058 
0.060 
0.062 
0.064 
0.066 

7.946 
8.152 
8.364 
8.582 
8.805 
9.035 

.206 

.212 

.218 
.223 
.230 

0.053 

0.055 
0.058 
O.o6o 
0.063 
O.o66 

i  103     0 

1598.82 

10.108 

0.068 

9.271 

0.069 

30 
104    0 

1620.75 
1642.91 

21.93 
22.16 

10.337 
10.570 

.229 

•233 

0.070 
0.072 

9-5I3 
9.761 

:^8 

.2«;6 

0.072 
0.075 

30 
105     0 

30 

1665.30 
1687.93 
1710.80 

22.39 
22.63 
22.87 

23.  12 

10.809 
11.053 
11.302 

•239 
.244 
.249 

0.074 
0.077 
0.079 

10.017 
10.280 
10.550 

.2% 
.270 
.278 

0.078 
O.o82 
0.085 

106    0 

30 
,  107    0 
30 
1  108     0 

30 

I733-92 
1757.28 
1780.90 
1804.77 
1828.90 
1853.30 

23.36 
23.62 
23.87 
24.13 
24.40 
24.67 

"•557 
11.817 

12.083 

12.354 
12.632 
12.916 

.260 
.266 
.271 
.278 
.284 
.291 

0.082 
0.084 
0.087 

0.093 

o  096 

10.828 
11.114 
1  1.408 
11.711 

12.022 

12-343 

.286 
.294 

•3°3 
.311 
.321 
.330 

0.089 
0.093 
0.098 
0.102 
O.IO7 
O.I  I  2 

109    0 

1877.97 

_f      i 

13.207 

0.099 

12.673 

O.II7 

615 


TABLE  IX. 

For  tindmfe  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  Orbits  of  great  eccentricity 


• 

A 

Diff. 

B 

Diff. 

c 

Diff. 

B' 

Diff. 

C" 

Diff. 

0       / 

n 

n 

n 

n 

„ 

11 

n 

II 

„ 

n 

109    0 

30 
110    0 
30 
111     0 

30 

1877.97 
1902.91 
1928.13 

1953.64 
1979.44 
2005.54 

24.94 
25.22 
25.51 
25.80 
26.10 
26.40 

13.207 
13.504 
13.808 
14.119 
14.438 
14.764 

•297 
-3°4 
•311 
.319 

.320 

-333 

0.099 

0.102 

0.106 
0.109 
0.113 
0.116 

.003 
.004 
.003 
.004 
.003 
.004 

12.673 
13.013 
13-363 
I3-724 
14.095 
14.478 

•34° 
.350 
.361 
.371 
.383 
.396 

0.117 

0.122 
0.128 
0.134 
0.141 
0.148 

,005 
.006 
.006 
.007 
.007 
.007 

112     0 
30 
113    0 

2031.94 
2058.64 
2085.66 

26.70 

27.02 

15.097 
15-439 
15-789 

.342 
•35° 

O.I  20 

0.124 

0.128 

.004 
.004 

14.874 
15.282 
15.702 

.408 
.420 

0.155 
O.l62 
0.170 

-.1:1 

30 
114    0 

30 

2113.00 
2140.66 
2168.66 

27-34 
27.66 
28.00 
28.34 

16.148 
16.515 
16.892 

•359 
.367 

:]ll 

0.132 

0.137 

0.142 

.004 
.005 
.005 
.005 

16.135 
16.583 
17.045 

•433 
.448 
.462 

-477 

0.178 
0.187 

o  196 

.008 

.009 

.009 

.010 

115    0 

30 
116    0 
3D 
117    0 

30 

2197.00 
2225.69 
2254.73 
2284.13 
2313.91 
2344.06 

28.69 
29.04 
29.40 
29.78 
3°-l5 
3°-54 

17.278 
17.674 
18.080 
18.496 
18.924 
19-363 

.396 

.406 
.416 
.428 
.439 

•45° 

0.147 

0.152 

0.157 

0.162 
0.168 
0.174 

.005 
.005 
.005 
.006 
.006 
.006 

17.522 
18.015 
18.524 
19.050 

19-594 

20.156 

-493 
•5°9 
.526 

-544 
.562 
.582 

0.206 
0.2  1  6 

0.227 

0.239 
0.251 

0.264 

.010 
.Oil 
.012 
.012 
.013 
.013 

118    0 

30 
119    0 
30 
120    0 

30 

2374.60 
2405.54 
2436.88 
2468.64 
2500.83 
2533-45 

3°-94 
31-34 
31.76 
32.19 
32.62 
33-o6 

19.813 

20.276 
20.751 
21.240 
21.742 
22.258 

•463 
•475 
•489 
.502 
.516 
•531 

0.180 
0.186 
0.193 

O.200 
0.207 
O.2I4 

.006 
.007 
.007 
.007 
.007 
.008 

20.738 

21-339 
21.962 
22.606 
23.273 
23.964 

.601 
.623 
.644 
.667 
.691 
.716 

0.277 

0.291 
0.306 
0.322 
0-339 
0-357 

.014 
.015 
.Ol6 
.017 
.018 
.019 

121     0 

30 
122     0 
30 
123    0 

30 

2566.51 
2600.03 
2634.02 
2668.49 
2703.46 
2738.93 

33-52 
3399 

34-47 
34-97 
35-47 
35-98 

22.789 
23.336 
23.898 
24.477 
25.073 
25.687 

-547 
.562 

•579 
.596 
.614 
•633 

0.222 
0.230 
0.239 
0.248 
0.258 
0.268 

.008 
.009 
.009 
.010 

.010 

.010 

24-680 
25.422 
26.191 
26.988 
27.815 
28.673 

.742 
.769 

-797 
.827 
.858 
.891 

0.376 
0.396 
0-417 
0-439 
0.463 
0.488 

.020 
.O2I 
.022 
.024 
.025 
.027 

124    0 

30 
125    0 
30 
126    0 

30 

2774.91 
2811.43 
2848.50 
2886.13 
2924.33 
2963.12 

36.52 

37.07 

37-63 
38.20 
38.79 
39-41 

26.320 
26.973 
27.646 
28.341 
29.057 

29.797 

.653 
.673 
.695 
.716 

.740 
-765 

0.278 
0.289 
0.300 
0.312 
0.325 
0.338 

.Oil 
.Oil 
.012 
.013 
.013 
.014 

29.564 
30.489 

3!-45o 
3M48 
33.485 
34-563 

.925 
.961 
.998 

•037 
.078 
.ri2 

0-5i5 

0.544 

0.606 
0.640 
0.676 

.029 
.030 
.032 
034 
.036 
.039 

127    0 

30 
128     0 
30 
129    0 

30 

3002.53 
3042.56 
3083.23 

31*4-57 
3166.59 
3209.31 

40.03 

40.67 

4L34 
42.02 
42.72 
43-45 

30.562 

31-351 

32.167 

33-on 
33.885 
34-789 

.789 
.816 
.844 
-874 
.904 

•936 

0-352 
0.367 
0.382 
0.398 
0-415 

0-433 

.015 
.015 
.Ol6 
.017 
.018 
.019 

35.685 
36.852 
38.067 

39-331 
40.649 
42.022 

.167 
.215 
.264 

.3!8 

-373 
.430 

0-715 
0-757 
0.800 
0.846 
0.896 
0.949 

.042 

.043 
.046 
.050 

-053 
.050 

130    0 

20 
40 
131     0 

20 
40 

3252.76 
3282.13 

3311-85 
3341.90 
3372.31 
3403.09 

29.37 
29.72 
30.05 
30.41 
30.78 
3i.!4 

35-725 
36.367 
37.025 
37.699 
38.389 
39.097 

.642 
.658 
.674 
.690 

.708 
.725 

0.452 
0.465 
0.479 

0-493 
0.508 
0.523 

.013 
.014 
.014 
.015 
.015 
.Ol6 

43.452 
44-439 
45-455 
46.500 

47-575 
48.682 

0.987 
.016 
.045 
•075 
.i°7 
.138 

.005 

•045 
.087 

•130 
•175 
.223 

.040 
.042 
•043 
-045 
.048 
.050 

132    0 

20 
40 
,  133    0 

20 
40 

3434-23 
3465.74 

3497-63 
3529.91 
3562.60 
3595-69 

31-51 
31-89 
32.28 

32-69 
33-09 
33-51 

39822 
40.564 
41-326 
42.108 
42.910 
43-733 

.742 
.762 
.782 
.802 
.823 
.843 

0.539 

0-555 
0-572 
0.590 
0.609 
0.629 

.Ol6 
.017 
.018 
.019 
.020 
.O2O 

49  820 
50.992 
52.199 

53-442 
54-723 
56.042 

.172 

.207 

:3? 

•319 
•359 

.273 
-325 
•379 
.436 
•495 
•558 

.052 

•054 
.057 
.059 
.063 
.005 

134    0 

20 
40 
135    0 

20 
40 

3629.20 
3663.13 
3697.50 
3732.31 
3767.58 
3803.31 

33-93 
34-37 
34.81 

35.27 
35-73 
36.21 

44.576 
45.442 

46-331 
47.245 
48.183 
49.147 

.866 
.889 
.914 

.964 
.991 

0.649 
0.669 
0.691 
o.7H 
0-738 
0763 

.020 
.022 
.023 
.024 
.025 
.025 

57.401 
58.802 
60.247 
61.736 
63.273 
64.857 

.401 

•445 
.489 

.537 

r4 
.634 

.623 
.692 
-764 
•839 
.917 

2.000 

.069 
.072 
.075 
.078 
.083 
.087 

136    0 

3839.52 

50.138 

0.788 

66.491 

2.087 

616 


TABLE  IX, 

f 01  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  Orbits  of  great  eccentricity 


X 

A 

Diff. 

B 

Diff. 

C 

Diff. 

B' 

Diff. 

C" 

Diff. 

0       1 

„ 

if 

„ 

n 

II 

n 

„ 

n 

„ 

„ 

136    0 

20 
40 
137    0 

20 

40 

3839-5* 
3876.21 
3913.41 
3951.12 
3989.35 
4028.11 

36.69 

37.20 

37-71 
38.23 
38.76 
39-31 

50.138 
51.156 
52.203 
53.280 
54.388 
55.528 

1.018 

1.047 
.077 
.108 
.140 
.174 

0.788 
0.815 
0.843 
0.873 
0.904 
0.936 

.027 
.028 
.030 
.031 
.032 
•°33 

66.491 
68.178 
69.920 
71.718 

73-575 
75-493 

.687 
•74* 
•798 
.857 
.918 
.982 

2.087 
2.178 
2.274 

*-375 
2.480 
2.591 

.091 
.096 
.101 

.105 
.Hi 
.117 

138    0 

20 
40 
139    0 

20 

40 

4067.42 
4107.28 
4147.72 
4188.75 
4230.38 
4272.63 

39-86 
40.44 
41.03 
41.63 
42.25 
42.89 

56.702 
57.910 
59-'54 
60-436 
61.757 
63.119 

.208 
.244 
.282 
.321 
.362 
.404 

0.969 
.004 
.041 

.079 
.119 

.161 

.035 

.040 
.042 
.044 

77-475 
79-5*3 
81.641 
81.830 
86.094 
88.436 

2.048 
2.118 
2.189 
2.264 
2.342 
2.424 

2.708 
2.831 
2.960 
3.096 
3.239 
3-39° 

.123 
.129  , 
.136 
.143 
.151 
.159 

140    0 

20 
40 
141     0 

20 
40 

4315.52 
4359.06 
4403.26 
4448.15 

4493-73 
4540.03 

43-54 
44.20 
44.89 

45.58 
46.30 
47.04 

64.523 
65.971 
67.465 
69.007 
70.599 
72.243 

.448 
•494 
•54* 
.592 
.644 
.698 

.205 
.251 
.299 

•350 
.404 

.460 

.046 
.048 
.051 
.054 
.056 
.058 

90.860 

93-369 
95.967 
98.657 
101.443 
104.331 

2.509 
2.598 
2.690 
2.786 
2.888 
2.993 

3-549 

3.717 

3-893 
4.080 
4.277 
4-484 

.168 
.176 
.187 
.197 
.207 

.220 

142     0 

10 
20 

30 
40 

50 

4587.07 
4610.88 
4634.88 
4659.07 
4683.46 
4708.05 

23.81 
24.00 
24.19 
24.39 
24.59 
24.79 

73-941 
74.811 

75-695 
76-595 
77.509 
78.439 

0.870 
0.884 
0.900 
0.914 
0.930 
0.946 

.518 

•549 
.580 
.612 
.645 
.679 

.031 
.031 

.032 
.033 
•°34 
•°35 

107.124 
108.861 
110.427 

I  I  2.  022 
113.646 
II5.30I 

•595 
.624 

•655 
.685 

4.704 
4.819 
4.936 

5-057 
5.181 
5.309 

.115 

.117 
.121 
.I2A 
.128 
.131 

143    0 

10 
20 
30 
40 
50 

4732.84 
4757.84 
4783.05 
4808.46 
4834.10 
4859-95 

25.00 
25.21 
25.41 
25.64 
25.85 
26.07 

79-385 
80.347 
81.325 
82.321 

83-333 
84.363 

0.962 
0.978 
0.996 

.012 
.030 
.048 

.714 

•749 
.786 
.823 
.862 
.901 

•035 
.037 
.037 

•°39 
.039 
.041 

116.986 
118.704 
120.452 
122.233 
124.049 
125.899 

.718 
.748 
.781 
.816 
.850 
.886 

5.440 
5-575 
5-7I5 
5-858 
6.005 
6.157 

.140 

•'43 

.147 

'\\l 

144    0 

10 
20 
30 
40 
50 

4886.02 
4912.31 
4938.81 
4965.58 
4992.56 
5019.78 

26.29 
26.52 
26.75 
26.98 
2.7.22 

85.411 

86.478 
87.564 
88.668 

89-793 
90.938 

.067 
.086 
.104 
.125 
.145 
.165 

•94* 
.984 
2.026 
2.070 
2.116 
2.162 

.042 
.042 
.044 
.046 
.046 

127.785 
129.707 
131.666 
133.661 
135.698 
137-774 

.922 
•959 
•997 
•035 
.076 
.116 

6.313 

6.473 
6.639 
6.809 
6.984 
7.165 

.l6o 
.166 

.170 

!i86 

145     0 

10 
20 
30 
40 

50 

5047.23 
5074.93 
5102.88 
5131.08 

5*59-53 
5188.24 

27.70 
27.95 
28.20 
28.45 
28.71 
28.97 

92.103 
93.290 
94.498 
95.729 
96.982 
98.259 

.187 
.208 
.231 

•*53 

•*77 
.300 

2.210 
2.259 
2.309 
2.361 
2.414 
2.469 

.049 

.050 
.052 

•053 
.055 
.057 

139.890 
142.048 
144.249 
146.494 
148.784 
151.120 

2.158 

2.2OI 
2.245 
2.290 
2.316 
2.383 

7-351 
7-543 
7.740 

8.'?533 

.192 
.197 
.203 

.210 
.216 
.223 

146    0 

10 
20 
30 
40 
50 

5217.21 
5246.45 
5*75-95 
53°5-73 
5335-79 
5366.13 

29.24 
29.50 
29.78 
30.06 

3°-34 
10.61 

99-559 

100.884 
102.234 
103.610 
105.012 
106.441 

•3*5 
•350 
•376 
.402 

•4*9 
.456 

2.526 

2.584 
2.643 
2.704 
2.767 
2.833 

.058 
.059 
.061 
.063 
.066 
.067 

I53-503 
155-934 
158.415 
160.947 
163.531 
I66.I68 

2.431 
2.4§I 
2.512 
2.584 
2.637 
2.692 

8.592 
8.822 
9.060 
9.304 

9-555 
9.815 

.230 
.238 
.244 

!268 

147     0 

10 
20 
30 
40 
50 

5396.76 
5427.67 
545888 
5490.39 
5522.20 
5554-33 

j      j 
30.91 
31.21 

3/.8i 
32.13 
32.44 

107.897 
109.182 
110.896 
112.439 
114.013 
115.619 

.485 
.514 

•543 

•637 

2.900 
2.969 
3.040 

3-IJ3 
3.188 
3.266 

.069 
.071 

•073 
.075 
.078 
.080 

168.860 
171.608 
174.414 
177.280 
180.206 
183.194 

2.748 
2.806 

2.866 
2.926 
2.988 
3.052 

10.083 
10.359 
10.645 
10.940 
11.244 
11.558 

.276 
.286  - 
.295 
.304 
.314 
•3*5 

148     0 

10 

20 
30 
40 
50 

149    0 

5586.77 
5619.52 
5652.60 
5686.01 
57I9-75 
5753-83 
5788.26 

33-o8 

33-74 
34.08 

34-43 

117.256 
118.926 
120.631 
122.370 
124.144 
125.955 

127.804 

.670 
.705 
•739 
•774 
.811 
.849 

3-346 
3-4*8 
3-5I3 
3.601 
3.691 

3.881 

.082 
.085 
.088 
.090 
.093 
.097 

186.246 
189.364 
192.549 
195.804 
199.130 
202.528 

2O6.OO2 

3.118 
3-185 
3-*55 
3-3*6 
3-398 
3-474 

11.883 
12.218 
12.564 
12.921 
13.291 
13.673 

14.067 

•335 
.346 

-357 
.370 
.382 
•394 

617 


TABLE  X. 

For  finding  the  True  Anomaly  or  the  Time  from  the  Perihelion  in  Elliptic  and  Hyperbolic  Orbits. 


A 

Ellipse. 

Hyperbola. 

log  B 

Diff. 

log  0 

log  I.  Diff. 

log 
half  II.  Diff. 

logJB 

Diff. 

logtf 

log  I.  Diff. 

log 
half  II.  Diff. 

0.000 

0.000 

o.oo 

oooo 

7 

0.000  OOOO 

4.23990 

1.778 

oooo 

o.ooo  oooo 

4.2  3  98  2n 

1.771 

.01 

0007 

/ 
27 

.001  7432 

.24286 

.783 

0007 

27 

9.998  2688 

.23686 

.767 

.02 

0030 

3 

77 

.003  4985 

.24583 

.788 

0030 

"3 

77 

.996  5493 

•23392 

.762 

.03 

0067 

31 

e-i 

.005  2659 

.24885 

•794 

0067 

3  / 
r  I 

.994  8414 

.23098 

•758 

.04 

0120 

68 

.007  0457 

.25190 

•799 

0118 

66 

•993  J45° 

.22807 

•753 

&.05 

0188 

84. 

0.008  8381 

4.25497 

1.805 

0184 

81 

9.991  4599 

4.225i8M 

1.748 

.06 

0272 

T 
00 

.010  6432 

.25806 

.811 

0265 

•989  7859 

.22230 

-743 

.07 

0371 

y  s 
I  14 

.012  4613 

.26116 

.816 

°359 

IOQ 

.988  1231 

.21943 

•739 

.08 
.09 

0485 

0615 

•  *-t 
130 

147 

.014  2924 

.016  1367 

.26427 
.26741 

.821 
.827 

0468 
0591 

*  W7 

123 

.986  4711 
.984  8298 

.21659 

.21376 

•734 
.730 

O.IO 

0762 

162 

0.017  9945 

4.27057 

I-833 

0728 

I  C2 

9.983  1992 

4-21094* 

1.725 

.11 

0924 

178 

.019  8659 

.27376 

•839 

0880 

I65 

.981  5791 

.20815 

.720 

.12 

1  102 

IQ4. 

.021  7511 

.27697 

•845 

1045 

•979  9694 

.20537 

.716 

•'3 

.14 

1296 
1507 

7T 

211 

227 

.023  6503 
.025  5637 

.28020 
.28344 

.851 
.857 

1223 
1416 

193 
2O6 

•978  3699 
.976  7805 

.20260 
.19986 

.706 

^ 

1734 
1977 

243 
26l 

0.027  49  r  6 
.029  4340 

4.28670 
.28999 

1.863 
.869 

1622 
1842 

220 

277 

9.975  201  1 

•973  6316 

4.19712,, 
.19440 

1.700 
•695 

.19 

2238 
2515 
2809 

277 
294 
3II 

•031  39*3 
.033  3636 
.035  3511 

.29331 
.29665 

.30001 

.875 
.882 
.888 

2075 
2321 
2581 

•v 

246 
260 
273 

.972  0719 
.970  5218 
.968  9813 

.19170 
.18901 
.18633 

.690 
.685 
.679 

O.2O 
.21 

3120 

3448 

328 

0.037  3542 
.039  3730 

4-3°339 
.30679 

1.895 
.901 

2854 
3140 

286 

9.967  4502 
.965  9285 

4.18367, 
.18102 

1.672 
.666 

.22 

3793 
4156 

363 
78l 

.041  4077 
.043  4585 

.31022 
.31368 

.908 
.915 

3439 
375i 

312 

.964  4159 
.962  9124 

.17840 
•17579 

.661 
.655 

.24 

4537 

J"  •* 
398 

.045  5259 

.31716 

.922 

4076 

338 

.961  4180 

.17319 

.649 

°'H 

4935 
5351 

4l6 

4.74. 

0.047  6099 
.049  7109 

4.32066 
.32418 

1.929 
•936 

4414 
4765 

767 

9.959  9324 
•958  4556 

.16803" 

1.643 
•637 

.27 

5785 

T  JT 
4.^2 

.051  8290 

•32773 

•943 

5128 

376 

•956  9875 

.16547 

.631 

.28 

6237 

TO  • 
4.71 

.053  9646 

.951 

55°4 

•955  5281 

.16292 

.625 

.29 

6708 

TV 
488 

.056  1179 

•33492 

.958 

5893 

401 

•954  °77i 

.16038 

.618 

0.30 

7196 

0.058  2893 

4.33856 

1.966 

6294 

9.952  6346 

4-i5785n 

1.613 

TABLE  X,    Part  II, 


r 

Ellipse. 

Hyperbola. 

T 

Ellipse. 

Hyperbola. 

A 

Diff. 

A 

Diff. 

A 

Diff. 

A 

Diff. 

o.oo 

.01 
.02 
.03 
.04 

O.OOOOO 

.00992 

.01969 
.02930 

.03877 

992 

977 
961 

947 

971 

o.ooooo 
.01008 

.02033 
.03074 
.04132 

1008 
1025 
1041 
1058 
1077 

0.20 
.21 

.22 

.24 

0.17266 
.18008 
.18740 
.19462 
.20174 

742 
732 
722 
712 
704 

0.23867 
.25309 
.26779 
.28280 
.29813 

1442 
1470 
1501 

1533 
1564 

0.05 
.06 

.09 

0.04808 

.05726 

.06630 

.07521 
.08398 

oo  rh  M  r^  w 
•H  O  ON  t^so 

ON  ONOO  OO  OO 

0.05209 

.06303 
.07417 

.08550 
.09702 

1094 
1114 

"33 
1152 

"73 

0.25 
.26 
.27 
.28 
.29 

0.20878 

•21573 
.22258 
.22935 
.23604 

695 

685 

677 
669 
661 

0.31377 

O.IO 

.11 

.12 
.I4 

0.09263 
.10116 
.10956 

.11783 
.12599 

853 

840 

827 

816 

805 

0.10875 
.12069 

.13285 
.14522 
.15782 

1194 
1216 
1237 
1260 
1285 

0.30 
32 

•33 
•34 

0.24265 
.24917 
.25561 
=26198 
.26826 

652 

644 

637 
628 
621 

O.I| 
.16 

•17 
.18 

0.13404 
.14198 

.14981 

•J5753 

794 

783 
772 

0.17067 

•18375 
.19709 
.21068 

1308 
1334 
1359 

T  1%f\ 

°| 

0.27447 
.28061 
.28668 
.29268 

614 

607 
600 

i   .19 

.16515 

762 
751 

.22454 

1413 

•39 

.29860 

592 
586 

0.20 

0.17266 

0.23867 

0.40 

0.30446 

618 


TABLE  XL 

For  the  Motion  in  a  Parabolic  Orbit. 


1? 

log/* 

Diff. 

, 

lOg  /A 

Diff. 

„ 

lOg/4 

Diff. 

0.000 

.001 

0.000  OOOO 
.OOO  OOOO 

0 

0.060 

.061 

o.ooo  0652 
.000  0674 

22 

0.120 
.121 

o.ooo  2617 
.000  2661 

44 

.002 

.000  0001 

I 

.062 

.000  0697 

23 

.122 

.000  2705 

44 

.oo3 

.000  0002 

I 
J 

.063 

.000  0719 

22 

.123 

.000  2750 

45 

.004 

.000  0003 

I 

.064 

.000  0742 

24 

.124 

.000  2795 

46* 

O.005 
.oo6 

o.ooo  0004 
.000  0006 

2 

0.065 

.066 

o.ooo  0766 
.000  0790 

24 

0.125 
.126 

o.ooo  2841 

.000  2886 

45 

.007 
.008 

.000  0009 

.000  0012 

3 
3 

.067 

.068 

.000  0814 
.000  0838 

24 

24 

.127 
.128 

.000  2933 

.OOO  2Q7Q 

46 

.009 

.000  0015 

3 
3 

.069 

.000  0863 

25 

.129 

S  I  7 
.OOO  3O26 

s 

O.OIO 

o.ooo  0018 

0.070 

o.ooo  0888 

0.130 

o.ooo  3074 

T° 

.Oil 

.OOO  OO22 

4 

.071 

.000  0914 

tft 

•'31 

.000  3121 

47 

.012 

.000  0026 

4 

5' 

.072 

.000  0940 

20 

.132 

.000  3169 

48 

.0!3 

.000  0031 

.073 

.000  0966 

20 

.000  3218    49 

.014 

.000  0035 

I 

.074 

.000  0993 

27 

27 

•'34 

.000  3267 

49 

49 

O.OI5 

o.ooo  0041 

0.075 

0.000  1020 

o  135 

o.ooo  3316 

.Ol6 

.000  0046 

I 

.076 

.000  1047 

27 

-0 

.136 

.000  3365 

49 

.017 

.000  0052 

.077 

.000  1075 

2o 
-0 

.137 

.000  3415 

5° 

.018 

.000  0059 

7 
5 

.078 

.000  1103 

2o 

.138 

.000  3466 

51 

.019 

.000  0065 

7 

.079 

.000  1132 

29 
29 

•'39 

.000  3516 

5° 
51 

0.020 

o.ooo  0072 

0.080 

o.ooo  1161 

0.140 

o.ooo  3567 

J 

.021 
.022 

.000  0080 
.000  0088 

8 

.081 
.082 

.000  1190 
.000  1219 

29 
29 

.141 

.142 

.000  3619 
.000  3671 

52 
5* 

.023 

.000  0096 

g 

.083 

.000  1249 

3° 

.000  3723 

52 

.024 

.000  0104 

9 

.084 

.000  1280 

31 
31 

.144 

.000  3775 

53 

0.025 

o.ooo  0113 

0.085 

o.ooo  1311 

0.145 

o.ooo  3828 

.026 

.000  0122 

9 

.086 

.000  1342 

31 

.146 

.000  3882 

54 

.027 

.000  0132 

IO 

.087 

.000  1373 

31 

.147 

.000  3935 

53 

.028 
.029 

.000  0142 
.000  0152 

IO 
10 

II 

.088 
.089 

.000  1405 
.000  1437 

32 
32 
33 

.148 
.149 

.000  3989 
.000  4044 

54 
55 

cc 

0.030 

o.ooo  0163 

I  I 

0.090 

o.ooo  1470 

A  -> 

0.150 

o.ooo  4099 

J  J 

.031 

.000  0174 

.091 

.000  1502 

32 

.151 

.000  4154 

55 

.032 

.000  0185 

1  1 
1  2 

.092 

.000  1536 

34 

0  -» 

.152 

.000  4209 

11 

•°33 

.000  0197 

•°93 

.000  1569 

33 

•J53 

.000  4265 

5° 

•°34 

.000  0209 

I  2r 

'3 

.094 

.000  1603 

34 
35 

.154 

.000  4322 

56 

0.035 

O.OOO  O222 

T  ^ 

0.095 

o.ooo  1638 

*  £ 

0.155 

o.ooo  4378 

.036 

.000  0235 

J3 

.096 

.000  1673 

35 

.156 

.000  4435 

-O 

•037 
.038 

.000  0248 
.000  0262 

14 

.097 
.098 

.000  1708 
.000  1743 

35 
35 

•'57 
.158 

.000  4493 
.000  4551 

5s 

$ 

.039 

.000  0275 

3 

.099 

.000  1779 

g 

.159 

.000  4609 

58 

15 

3° 

59 

0.040 

o.ooo  0290 

0.100 

o.ooo  1815 

0.160 

o.ooo  4668 

.0 

.041 

.000  0304 

J4 

I£ 

.101 

.000  1852 

37 

.161 

.000  4726 

r 

.042 

.000  0320 

o 

.102 

.000  1889 

37 

.162 

.000  4786 

Oo 

(\r\ 

.043 

.000  0335 

*5 

.103 

.000  1926 

11 

.163 

.000  4846 

OO 
fin 

.044 

.000  0351 

I  O 

16 

.104 

.000  1964 

3° 

38 

.164 

.000  4906 

OO 

60 

0.045 
.046 
.047 

o.ooo  0367 
.000  0383 
.000  0400 

16 

17 

0.105 
.106 

.107 

0.000  2002 

.000  2040 
.000  2079 

3« 
39 

0.165 
.166 
.167 

o.ooo  4966 
.000  5027 
.000  5088 

61  I 
61 

.048 

.000  0417 

•  Q 

.108 

.000  21  1  8 

39 

.168 

.000  5150 

ft 

.049 

.000  0435 

15 

18 

.109 

.000  2158 

4° 
4° 

.169 

.000  5212 

62 

0.050 

o.ooo  0453 

•I 

O.IIO 

o.ooo  2198 

0.170 

o.ooo  5274 

fi 

.051 

.000  0471 

I  o 

.III 

.000  2238 

4° 

.171 

.000  5337 

?3 

.052 

•°53 
.054 

.000  0490 
.000  0509 
.000  0528 

19 
19 

20 

.112 

.113 
.114 

.ooc  2279 
.000  2320 
.000  2361 

H  IH  M  f< 

.172 
.173 
.174 

.000  5400 
.000  5464 
.000  5518 

i* 

64 

64 

0.055 

o.ooo  0548 

0.115 

o.ooo  2403 

0.175 

o.ooo  5592 

, 

.056 

.000  0568 

20 

.116 

.000  2445 

42 

.176 

.000  5657 

£ 

•°57 

.000  0589 

21 

.117 

.000  2487 

42 

.177 

.000  5722 

ftC 

.000  0610 

21 

.118 

.000  2530 

43 

.178 

.000  5787 

05 

ftfi 

.059 

.000  0631 

21 
21 

.119 

.000  2573 

43 
44 

.179 

.000  5853 

OO 

66 

0.060 

o.ooo  0652 

0.120 

o.ooo  2617 

0.180 

o.ooo  5919 

619 


TABLE  XI. 

For  the  Motion  in  a  Parabolic  Orbit. 


„ 

log/* 

Diff. 

•n 

>og. 

Diff. 

, 

*» 

Diff. 

o.i8o 
.181 

o.ooo  5919 
.000  5986 

6? 

0.240 
.241 

o.ooi  0603 
.001  0693 

90 

0.300 
.301 

o.ooi  6733 
.001  6848 

"5 

.182 

.000  6053 

07 

.242 

.001  0784 

91 

.302 

.001  6963 

115 

1  1  6 

.183 
.184 

..ooo  6120 

.000  6l88 

68 
68 

•243 
.244 

.001  0875 
.001  0966 

91 

91 

92 

•3°3 
.304 

.001  7079 
.001  7195 

116 
117 

0.185 
.186 
.187 
.188 
.189 

o.ooo  6256 
.000  6325 
.000  6393 
.000  6463 
.000  6532 

69 

68 

7° 
69 

7° 

0.245 
.246 
.247 
.248 
.249 

o.ooi  1058 
.001  1150 
.001  1242 
.001  1335 
.001  1429 

92 
92 

93 
94 
93 

0.305 
.306 

•3°7 
.308 
.309 

o.ooi  7312 
.001  7429 
.001  7546 
.001  7664 
.001  7783 

117 
117 
118 
119 
118 

0.190 

o.ooo  6602 

0.250 

o.ooi  1522 

n  e 

0.310 

o.ooi  7901 

.191 
.192 

•193 
.194 

.000  6673 
.000  6744 
.000  6815 
.000  6887 

72 
72 

.251 
.252 

•253 
.254 

.001  1617 
.001  1711 
.001  1806 
.001  1901 

95 

94 
95' 

96 

•3" 

.312 

•3'3 
.314 

.001  8020 
.001  8140 
.001  8260 
.001  8381 

119 

120 
120 
121 
121 

0.195 

o.ooo  6959 

0.255 

o.ooi  1997 

06 

°-3'5 

o.ooi  8502 

121 

.196 

.000  7031 

mm 

.256 

.obi  2093 

yu 

.316 

.001  8623 

I  22 

.197 
.198 
.199 

.000  7104 
.000  7177 
.000  7250 

73 

73 
73 
74 

.257 
.258 
.259 

.001  2190 
.001  2287 
.001  2384 

97 
97 
98 

.317 
.318 
.319 

.001  8745 
.001  8867 
.001  8989 

122 

122 
124 

0.200 
.201 

o.ooo  7324 
.000  7399 

75 

0.260 
.261 

o.ooi  2482 
.001  2580 

98 

0.320 
.321 

o.ooi  9113 
.001  9236 

123 

.202 

.000  7473 

74 

.262 

.001  2679 

99 

.322 

.001  9360 

124 

.203 

.000  7548 

11 

.263 

.001  2778 

99 

.001  9484 

124 

.204 

.000  7624 

70 
76 

.264 

.001  2877 

99 

100 

•3*4 

.001  9609 

125 
125 

O.205 

o.ooo  7700 

76 

0.265 

o.ooi  2977 

IOO 

0.325 

o.ooi  9734 

126 

.206 
.207 
.208 

.000  7776 
.000  7853 
.000  7930 

/  u 
77 
77 

*7*t 

.266 
.267 
.268 

.001  3077 
.001  3178 
.001  3279 

101 
101 

I  02 

.326 

•3*7 
.328 

.001  9860 
.001  9986 

.002  0113 

126 

127 

.209 

.000  8007 

78 

.269 

.001  3381 

101 

•3*9 

.002  0240 

127 

O.2IO 
.211 
.212 

o.ooo  8085 
.000  8163 
.000  8242 

78 
79 

0.270 
.271 

.272 

o.ooi  3482 
.001  3585 

.001  3688 

103 
103 

T  Of 

0.330 
•331 
•33* 

O.OO2  0367 
.002  0495 
.002  0624 

128 
129 

.213 
.214 

.000  8321 
.000  8400 

79 

11 

.273 
.274 

.001  3791 
.001  3894 

103 
103 

T  r*A 

•333 
•334 

.002  0752 
.002  0882 

130 

oO 

104 

129 

O.2I5 
.216 

o.ooo  8480 
.000  8560 

80 

0.275 
.276 

o.ooi  3998 
.001  4103 

I05 
T  C\A 

0-335 
.336 

0.002  1  01  1 
.002  1141 

I30 

.217 

.000  8641 

0T 

.277 

.001  4207 

104 

•337 

.002  1272 

131 

.218 

.000  8722 

o  I 

81 

.278 

.001  4313 

.338 

.002  1403 

131 

.219 

.000  8803 

o  I 

82 

.279 

.001  4418 

"I 

•339 

.002  1534 

132 

O.220 
.221 

.222 

o.ooo  8885 
.000  8967 
.000  9050 

82 

ll 

0.280 
.281 
.282 

o.ooi  4524 
.001  4631 
.001  4738 

107 
107 

T  0*7 

0.340 
.341 
.342 

O.OO2  l666 
.002  1799 
.002  1971 

133 
132 

T  *»  A 

.223 

.224 

.000  9132 
.000  9216 

oZ 

84 
84 

.283 
.284 

.001  4845 
.001  4953 

107 
108 
108 

•343 
•344 

.002  2065 
.002  2198 

1  34 

133 

0.225 
.226 
.227 
.228 

o.ooo  9300 
.000  9384 
.000  9468 
.000  9553 

84 
84 

5< 

0.285 
.286 
.287 
.288 

o.ooi  5061 
.001  5169 
.001  5278 
.001  5388 

1  08 
109 
no 

o-345 
•346 
•347 
•348 

0.002  2333 
.OO2  2467 
.002  2602 
.002  2738 

134 

'35 

136 

.229 

.000  9638 

ll 

.289 

.001  5497 

109 
in 

•349 

.002  2874 

l\6 

O.230 

o.ooo  9724 

86 

0.290 

o.ooi  5608 

I  JO 

0.350 

0.002  3010 

.231 

.232 
.233 
.234 

.000  9810 
.000  9897 
.000  9984 
.001  0071 

3 

ll 

.291 
.292 
.293 
.294 

.001  5718 
.001  5829 
.001  5941 
.001  6053 

in 

112 
112 
112 

•351 
•352 
•353 
•354 

.002  3147 
.002  3284 
.OO2  3422 
.002  3560 

137 
138 
138 
139 

0.235 
.236 
.237 

o.ooi  0159 
.001  0247 
.001  0335 

88 
88 

0.295 
.296 
.297 

o.ooi  6165 
.001  6278 
.001  6391 

II3 
"3 

*$ 

•357 

0.002  3699 
.002  3838 

.002  3977 

139 
139 

.238 
.239 

.001  0424 
.001  0513 

89 
9° 

.298 
.299 

.001  6505 
.001  6619 

II4 
114 

•358 
•359 

.002  4117 

.002  4258 

141 
141 

0.240 

o.ooi  0603 

0.300 

o.ooi  6733 

0.360 

0.002  4399 

620 


TABLE  XI. 

For  the  Motion  in  a  Parabolic  Orbit. 


, 

log/* 

Diff. 

, 

*, 

Diff. 

, 

log  ft 

Diff. 

o  «  n  co  tj- 

so  so  so  so  so 

CO  CO  CO  to  CO 

d  • 

0.002  4399 

.002  4540 
.002  4682 
.002  4824 
.002  4967 

141 

142 
142 
143 

143 

0.420 

.421 

.422 

•4*3 

.424 

0.003  3720 
.003  3890 
.003  4061 
.003  4232 
.003  4404 

170 
171 
171 

I72 
172 

0.480 
.48l 
.482 
•483 
•484 

0.004  4858 
.004  5061 
.004  5263 
.004  5467 
.004  5670 

203 

202 
204 
203 
205 

0.365 

.366 

O.002  5IIO 
.002  5254 

144 

0.425 

.426 

0.003  4576 
.003  4749 

173 

0.485 
.486 

0.004  5875 
.004  6080 

205 

.367 

•368 
.369 

.002  5398 

.002  5543 

.002  5688 

144 
'45 
'45 
146 

.427 

.428 
.429 

.003  4923 
.003  5096 
.003  5271 

173 
175 
J74 

.487 
.488 

•489 

.004  6285 
.004  6492 
.004  6698 

205 
20J 
206 
208 

0.370 

0.002  5834 

0.430 

0.003  5445 

0.490 

0.004  6906 

•371 
.372 

.002  5980 
.002  6l26 

146 

•43  1 

.432 

.003  5621 
.003  5797 

!76 

f  nf\ 

.491 
.492 

.004  7113 
.004  7322 

207 
209 

•373 

.002  6273 

Hg 

•433 

•003  5973 

170 

•493 

.004  7531 

209 

•374 

.002  6421 

•434 

.003  6150 

177 

177 

•494 

.004  7740 

209 
211 

0-375 
•376 

0.002  6568 
.002  6717 

149 

y  A  f\ 

o-435 
.436 

0.003  6327 
.003  6505 

I78 

1  7% 

o-495 
•496 

0.004  7951 
.004  8161 

210 

•377 
.378 

.002  6866 
.002  7015 

149 
149 

»  rr\ 

•437 
•438 

.003  6683 
.003  6862 

178 
III 

•497 
•498 

.004  8373 
.004  8585 

212 

•379 

.002  7165 

150 

150 

•439 

.003  7042 

I  oO 

180 

•499 

.004  8797 

212 

213 

O  M  N  COrJ- 
OO  00  00  00  00 
CO  CO  CO  CO  CO 

d 

0.002  7315 
,OO2  7466 
.002  7617 
.OO2  7769 
.002  7921 

151 
151 
152 
152 

152 

0.440 
.441 
.442 
•443 
•444 

0.003  722a 
.003  7402 
.003  7583 
.003  7765 
.003  7947 

180 
181 

182 
183 

0.500 

•51 

.52 

•53 
•54 

0.004  9010 
.005  1173 
.005  3397 
.005  5681 
.005  8029 

2163 
2224 
2284 
2348 
2412 

0.385 
.3^6 

•387 
•388 

•389 

0.002  8073 
.002  8226 
.002  8380 
.002  8574 
.002  8689 

'53 

154 
'55 

155 

o-445 
•446 
•447 
.448 
•449 

0.003  8130 
.003  8313 
.003  8496 
.003  8680 
.003  8865 

183 
183 
184 
185 
185 

•59 

0.006  0441 
.006  2919 
.006  5464 
.006  8079 
.007  0765 

2478 

*545 
2615 
2686 
2760 

0.390 
.391 

0.002  8844 
.OO2  8999 

J55 

0.450 
•451 

0.003  9°5° 
.003  9236 

186 
186 

0.60 
.61 

0.007  3525 
.007  6361 

2836 

201  1 

•39* 
•393 
•394 

.002  9155 
.002  9311 
.002  9468 

^56 
157 
158 

.452 
•453 
•454 

.003  9422 
.003  9609 
•003  9797 

187 
188 
187 

.62 

I3 

.64 

.007  9274 
.008  2268 
.008  5345 

*7  5 

2994 

3°77 
3163 

o-395 

•396 
•397 
•398 

O.OO2  9626 

.002  97g4 

.002  9942 

.003  oioi 

158 
158 
i59 

0-455 
•45  6 
•457 
•458 

0.003  9984 
.004  0173 
.004  0362 
.004  0551 

189 
189 
189 

0.65 
.66 
•67 
.68 

0.008  8508 
.009  1759 
.009  5103 
.009  8542 

3*51 

3344 
3439 

3C  1  O 

•399 

.003  0260 

*59 
160 

•459 

.004  0741 

191 

.69 

.010  2081 

33V 

3642 

0.400 

0.003  0420 

0.460 

0.004  °9  3  2 

0.70 

o.oio  5723 

17CO 

.401 
.402 

•4°3 
.404 

.003  0580 
.003  0741 
.003  0903 
.003  1064 

161 
162 
161 
163 

.461 
.462 
.463 
.464 

.004  1123 
.004  1315 
.004  1507 
.004  1700 

192 
192 
193 
193 

.72 
•73 
•74 

.010  9473 
.on  3336 
.on  7316 
.012  1419 

•*£-> 
3863 
3980 
4103 

4*33 

0.405 
.406 
.407 
.408 
.409 

0.003  1227 
.003  1389 

.003  1553 

.003  1716 
.003  1881 

162 
164 
163 

'65 

164 

[467 
.468 
.469 

0.004  1893 
.004  2087 
.004  2281 
.004  2476 
.004  2672 

194 
194 

'95 
196 
196 

0.75 
.76 

•77 
.78 

•79 

0.012  5652 
.013  0022 
.013  4536 
.013  9202 
.014  4031 

4370 

45  H 
4666 
4829 
5002 

0.410 
.411 

.412 

•413 
.414 

0.003  2045 

.003  221  I 

.003  2376 
.003  2543 

.003  2709 

166 
165 
167 
166 
168 

0.470 
.471 
.472 
•473 
•474 

0.004  2868 
.004  3064 
.004  3261 
.004  3459 
.004  3657 

196 
197 
198 
198 
199 

0.80 
.81 
.82 

0.014  9033 
.015  4219 
.015  9603 
.Ol6  5202 
.017  1033 

5186 

5384 
5599  ' 
5831  | 
6087 

0.415 
.416 
.417 
.418 
.419 

0.003  ^877 

.003  3044 
.003  3213 
.003  3381 
.003  3550 

167 
169 
168 
169 
170 

0-475 
•476 
•477 
•478 
•479 

0.004  3856 
.004  4055 
.004  4255 
.004  4456 
.004  4657 

199 

200 

201 
201 
201 

0.85 
.86 

•87 
.88 
.89 

O.OI7  7I10 
.Ol8  3486 
.019  0165 
.019  7195 
.020  4629 

6366 
6679 
7030 

7434 
7900 

0.420 

0.003  3720 

0.480 

0.004  4858 

0.90 

0.021  2529 

621 


TABLE  XII, 


9 

t 

if 

i 

•  . 

.4 

3' 

4 

A 

S 

log  wij 

log  TO2 

ml 

m, 

m, 

TO! 

™i 

m3 

7H2 

mi 

—  0  0 

00 

0.0000 

0   0 

90  o 

90  o 

180  o 

180  o 

180  o 

0   0 

0   0 

1 

4.2976 

9.9999 

2  23 

90  20 

90  20 

178  40 

178  40 

179  o 

359  o 

359  5' 

2 

3-395° 

9.9996 

446 

90  40 

90  40 

177  20 

177  20 

178  o 

358  o 

358  9 

3 

2.8675 

9.9992 

7  8 

91  o 

91  o 

176  o 

176  o 

177  o 

357  o 

357  H 

4 

2.4938 

9.9986 

9  32 

91  20 

91  20 

174  40 

174  40 

176  o 

356  o 

356  18 

5 

2.2044 

9.9978 

"  55 

9I  41 

91  41 

173  19 

173  19 

175  o 

355  ° 

355  Z3 

6 

.9686 

9.9968 

14  19 

92   I 

92   I 

171  59 

171  59 

174  o 

354  o 

354  28 

7 

.7698 

9-9957 

1  6  42 

92  22 

92  22 

170  38 

170  38 

172  59 

353  * 

353  3^ 

8 

.S98i 

9-9943 

19  7 

92  42 

92  42 

169  18 

169  18 

171  59 

35*  i 

352  37, 

9 

•4473 

9.9928 

ai  32 

93  3 

93  3 

167  57 

167  57 

170  58 

35i  2 

351  4a; 

10 

.3130 

9.9911 

23  57 

93  25 

93  *5 

166  35 

166  35 

169  57 

35°  3 

35°  47! 

11 

.1922 

9.9892 

26  23 

93  46 

93  46 

165  14 

165  14 

1  68  55 

349  4 

349  52; 

12 

.0824 

9.9871 

28  50 

94  8 

94  8 

163  52 

163  52 

167  54 

348  6 

348  56: 

13 

0.9821 

9.9848 

31  i? 

94  31 

94  3i 

162  29 

162  29 

166  51 

347  8 

348  i 

14 

0.8898 

9.9823 

33  46 

94  53 

94  53 

161  7 

161  7 

165  48 

346  ii 

347  6: 

15 

0.8045 

9.9796 

36  15 

95  *7 

95  '7 

'59  43 

'59  43 

164  44 

345  H 

346  ii 

16 

0.7254 

9.9767 

38  46 

95  40 

95  4° 

158  20 

158  20 

163  40 

344  17 

345  '6 

17 

0.6518 

9.9736 

41  18 

96  5 

96  5 

156  55 

156  55 

162  34 

343  ai 

344  21 

18 

0.5830 

9.9702 

43  5i 

96  30 

96  30 

155  30 

'55  3° 

161  27 

342  27 

343  27 

19 

0.5185 

9.9667 

46  26 

96  56 

96  56 

154  4 

154  4 

160  19 

341  32 

342  32 

20 

0.4581 

9.9629 

49  2 

97  23 

97  a3 

"5a  37 

'Sa  37 

'59  9 

340  38 

34i  37 

21 

0.4013 

9.9588 

51  41 

97  5° 

97  5° 

151  10 

151  10 

157  58 

339  45 

34°  43> 

22 

0.3479 

9-9545 

54  »» 

98  19 

98  19 

149  41 

149  41 

156  45 

338  53 

339  49 

23 

0.2976 

9-9499 

57  5 

98  49 

98  49 

148  ii 

148  ii 

155  29 

338  o 

338  54| 

24 

0.2501 

9-945  1 

59  51 

99  20 

99  20 

146  40 

146  40 

154  ii 

337  9 

338  o 

25 

0.2053 

9.9400 

62  40 

99  53 

99  53 

H5  7 

H5  7 

152  50 

336  19 

337  6' 

26 

0.1631 

9-9345 

65  33 

100  28 

100  28 

143  32 

H3  32 

I51  25 

335  28 

336  13 

27 

0.1232 

9.9287 

68  30 

ioi  5 

ioi  5 

'4i  55 

HI  55 

149  56 

334  38 

335  '9 

28 

0.0857 

9.9226 

7i  33 

101  45 

ioi  45 

140  15 

140  15 

148  22 

333  49 

334  25 

29 

0.0503 

9.9161 

74  4i 

102  27 

102  27 

138  33 

138  33 

146  42 

333  i 

333  32 

30 

0.0170 

9.9092 

77  58 

I03  I3 

I03  I3 

136  46 

136  46 

'44  55 

332  12 

33*  39 

31 

9-9857 

9.9019 

8x  23 

104  4 

104  4 

134  56 

134  56 

142  59 

331  24 

331  46 

32 

9.9565 

9.8940 

85  o 

105  i 

105  i 

'32  59 

'3*  59 

140  51 

33°  37 

330  54 

33 

9.9292 

9.8856 

88  54 

106  6 

106  6 

'3°  54 

13°  54 

138  27 

329  49 

33°  2 

34 

9.9040 

9-8765 

93  " 

IO7  22 

107  22 

128  38 

128  38 

135  39 

329  2 

329  10 

35 

9.8808 

9.8665 

98  7 

108  58 

108  58 

126   2 

126   2 

132  13 

328  14 

328  19 

36 

9.8600 

9-8555 

104  20 

III  13 

III  13 

122  47 

122  47 

127  29 

327  27 

327  28 

—36  52.2 

9-8443 

9.8443 

116  34 

116  34 

116  34 

116  34 

116  34 

116  34 

3*6  45 

3^6  45 

This  table  exhibits  the  limits  of  the  roots  of  the  equation 

sin  (/  —  C)  =  mo  sin4  z', 

when  there  are  four  real  roots.  The  quantities  ml  and  w2  are  the  limiting 
values  of  ra0,  and  the  values  of  z/,  z2'9  z/,  and  z/,  corresponding  to  ea(;h  of 
these,  give  the  limits  of  the  four  real  roots  of  the  equation. 


622 


TABLE  XII, 


Y 

z 

i' 

z 

/ 

1 

8 

* 

/ 

4 

b 

log  TO! 

log  ?M£ 

m, 

m, 

ml 

ro, 

n»2 

«! 

mi 

m3 

+  00 

00 

0.0000 

0   0 

0   0 

O   0 

90  o 

90  0 

180  o 

180  o 

180  o 

1 

4.2976 

9-9999 

I   0 

I  20 

I  20 

89  40 

89  40 

177  37 

180  55 

181  o! 

2 

3-395° 

9.9996 

2   O 

2  40 

2  40 

89  20 

89  20 

175  14 

181  51 

182  o' 

8 

2.8675 

9.9992 

3  o 

4  o 

4  o 

89  o 

89  o 

172  52 

182  46 

183  o 

4 

2.4938 

9.9986 

4  o 

5  20 

5  20 

88  40 

88  40 

170  28 

183  4* 

184  o 

5 

2.2044 

9.9978 

5  o 

6  41 

6  41 

88  19 

88  19 

168  5 

184  37 

185  o1 

6 

.9686 

9.9968 

6  o 

8  i 

8  i 

87  59 

87  59 

165  41 

185  32 

186  o 

7 

.7698 

9-9957 

7  i 

9  22 

9  22 

87  38 

87  38 

163  18 

186  28 

186  59, 

8 

.5981 

9-9943 

8  i 

10  42 

10  42 

87  18 

87  18 

160  53 

187  23 

187  59 

9 

•4473 

9.9928 

9  * 

i*  3 

12  3 

86  57 

86  57 

158  28 

188  18 

188  58! 

10 

.3130 

9.9911 

10  3 

13  25 

13  25 

86  35 

86  35 

i56  3 

189  13 

189  57 

11 

.1922 

9.9892 

ii  5 

14  46 

14  46 

86  14 

86  14 

153  37 

190  8 

19°  56: 

12 

.0824 

9.9871 

12   6 

16  8 

16  8 

85  52 

85  52 

151  10 

191  4 

191  54; 

13 

0.9821 

9.9848 

13  9 

17  31 

17  31 

85  29 

85  29 

148  43 

191  59 

192  52 

14 

0.8898 

9.9823 

14  12 

18  53 

18  53 

85  7 

85  7 

146  14 

192  54 

193  49 

15 

0.8045 

9.9796 

15  16 

20  17 

20  17 

84  43 

84  43 

H3  45 

193  49 

194  46 

16 

0.7254 

9.9767 

I  6  20 

21  40 

21  40 

84  20 

84  20 

141  H 

194  44 

195  43, 

17 

0.6518 

9.9736 

17  26 

^3   5 

23  5 

83  55 

83  55 

138  42 

195  39 

196  39J 

18 

0.5830 

9.9702 

18  33 

24  30 

24  30 

83  3° 

83  30 

136  9 

196  33 

197  33i 

11) 

0.5185 

9.9667 

19  41 

25  56 

25  56 

83  4 

83  4 

133  34 

197  28 

198  28 

20 

0.4581 

9.9629 

20  51 

27  23 

27  23 

82  37 

82  37 

130  58 

198  23 

199  22 

21 

0.4013 

9.9588 

22   2 

28  50 

28  50 

82  10 

82  10 

128  19 

199  17 

200  15 

22 

0-3479 

9-9545 

23  15 

30  19 

30  19 

81  41 

8  1  41 

125  38 

200  II 

201  07 

23 

0.2976 

9.9499 

24  31 

3i  49 

3i  49 

81  ii 

81  ii 

122  55 

201   6 

202   0 

24 

0.2501 

9-945  i 

25  49 

33  ^o 

33  20 

80  40 

80  40 

120  9 

202   0 

202  51 

25 

0.2053 

9.9400 

27  10 

34  53 

34  53 

80  7 

80  7 

117  20 

202  54 

203  41 

26 

0.1631 

9-9345 

28  35 

36  28 

36  28 

79  32 

79  32 

114  27 

203  47 

204  32 

27 

0.1232 

9.9287 

3°  4 

38  5 

38  5 

78  55 

78  55 

III  30 

204  41 

205  22 

1  28 

0.0857 

9.9226 

3i  38 

39  45 

39  45 

78  15 

78  15 

108  27 

205  35 

206  II 

29 

0.0503 

9.9161 

33  18 

4i  27 

4i  27 

77  33 

77  33 

i°5  19 

206  28 

206  59 

30 

0.0170 

9.9092 

35  5 

43  13 

43  n 

76  47 

76  47 

102  3 

207  a  i 

207  48 

1  31 

9-9857 

9.9019 

37  i 

45  4 

45  4 

75  56 

75  56 

98  37 

208  14 

208  36 

32 

9.9565 

9.8940 

39  9 

47  * 

47  i 

74  59 

74  59 

95  ° 

209  06 

209  23 

33 

9.9292 

9.8856 

4i  33 

49  6 

49  6 

73  54 

73  54 

91  6 

209  58 

210  II 

34 

9.9040 

9.8765 

44  21 

5I  22 

51  22 

72  38 

72  38 

86  49 

210  50 

210  58 

35 

9.8808 

9.8665 

47  47 

53  58 

53  58 

71   2 

71  2 

81  53 

211  41 

211  46 

36 

9.8600 

9.8555 

52  3i 

57  13 

57  13 

68  47 

68  47 

75  40 

212  32 

212  33 

+36  52.2 

9-8443 

9.8443 

63  26 

63  26 

63  26 

63  26 

63  26 

63  26 

213  15 

2I3  15 

This  table  exhibits  the  limits  of  the  roots  of  the  equation 

sin  (d  —  C)  =  m0  sin*  2', 

when  there  are  four  real  roots.  The  quantities  ml  and  m^  are  the  limiting 
values  of  m0,  and  the  values  of  z/>  zz'>  zs>  and  z*>  corresponding  to  each  of 
these,  give  the  limits  of  the  four  real  roots  of  the  equation. 


623 


TABLE  XIII, 

For  finding  the  Ratio  of  the  Sector  to  the  Triangle. 


•n 

log*S 

Diff. 

• 

logs* 

Diff. 

i 

to* 

Diff. 

o.oooo 

.0001 

.0002 
.0003 
.0004 

O.OOO  0000 

.000  0965 
.000  1930 
.000  2894 
.000  3858 

965 
965 
964 

964 
963 

0.0060 
.0061 
.0062 
.0063 
.0064 

0.005  7*98 

.005  8243 

.005  9187 
.006  0131 
.006  1075 

945 

944 
944 
944 
944 

0.0120 
.OI2I 
.0122 
.0123 
.0124 

o.on  3417 
.on  4343 
.on  5268 
.on  6193 
.on  7118 

926 
925 
925 
925 
925 

0.0005 

o.ooo  4821 

0.0065 

0.006  2019 

O.OI25 

o.on  8043 

.0000 

.0007 
.0008 

.0009 

.000  5784 
.000  6747 
.000  7710 
.000  8672 

963 
963 

963 
962 
962 

.0066 
.0067 
.0068 
.0069 

.006  2962 
.006  3905 

.006  4847 
.006  5790 

943 

943 
942 

943 
942 

.0126 
.0127 
.0128 
.0129 

.on  8967 
.01  1  9890 

.012  0814 
.012  1737 

924  ! 
923 
924 
923 
923 

O.OOI  0 

.001  1 

.0012 

.0013 
.0014 

o.ooo  9634 
.001  0595 
.001  1556 
.001  2517 
.001  3478 

961 
961 
961 
961 
960 

0.0070 
.0071 
.0072 
.0073 
.0074 

0.006  6732 

.006  7673 
.006  8614 
.006  9555 
.007  0496 

941 
941 
941 
941 
940 

0.0130 
.0131 
.0132 
.0133 
.0134 

0.012  2660 
.012  3583 
.012  4505 
.012  5427 
.012  6348 

9*3 
922 
922 
921 
921 

0.0015 
.0016 
.0017 

o.ooi  4438 
.001  5398 
.001  6357 

960 
959 

0.0075 
.0076 
.0077 

0.007  *436 
.007  2376 
.007  3316 

940 

94° 

0.0135 
.0136 
.0137 

0.012  7269 
.012  8190 
.012  9III 

921 

921  i 

.0018 
.0019 

.001  7316 
.001  8275 

959 
959 

.0078 
.0079 

.007  4255 
.007  5194 

939 
939 

n^  n 

.0138 
.0139 

.013  0032 
.013  0952 

921 
920 

n  T  n 

959 

y3:/ 

y*y 

O.OO2O 
.0021 
.0022 
.0023 
.0024 

o.ooi  9234 

.002  0192 
.002  1150 
.002  2107 
.002  3064 

958 
958 
957 
957 
957 

0.0080 
.0081 
.0082 
.0083 
.0084 

0.007  6133 
.007  7071 
.007  8009 
.007  8947 
.007  9884 

938 
938 
938 

937 
937 

0.0140 
.0141 
.0142 
.0143 
.0144 

0.013  I^7I 
.013  2791 
.013  3710 
.013  4629 
.013  5547 

920 
919 

9*9  ! 
918 
918 

O.0025 
.0026 
.0027 
.O028 
.OO29 

0.002  4021 

.002  4977 
.002  5933 

.002  6889 
.002  7845 

956 
956 
956 
956 

955 

0.0085 
.0086 
.0087 
.0088 
.0089 

0.008  0821 
.008  1758 
.008  2694 
.008  3630 
.008  4566 

937 
936 
936 
936 
936 

0.0145 
.0146 
.0147 
.0148 
.0149 

0.013  6465 
.013  7383 
.013  8301 
.013  9218 
.014  0135 

918 
918 
917 
917 
917 

O.0030 
.0031 
.0032 
•0033 
.0034 

0.002  8800 

.002  9755 
.003  0709 
.003  1663 
.003  2617 

955 
954 
954 
954 
953 

0.0090 
.0091 
.0092 
.0093 
.0094 

0.008  5502 
.008  6437 
.008  7372 
.008  8306 
.008  9240 

935 
935 
934 
934 
934 

0.0150 
.0151 
.0152 

•°153 
.0154 

0.014  I052 
.014  1968 
.014  2884 
.014  3800 
.014  4716 

916 
916 
916 
916 

0.0035 

0.003  3570 

n  f  * 

0.0095 

0.009  OI74 

O  -5  A 

0.0155 

0.014  563! 

.0036 
.0037 
.0038 
.0039 

.003  4523 
.003  5476 

.003  6428 
.003  7380 

y.>3 
953 
952 
952 
952 

.0096 
.0097 
.0098 
.0099 

.009  i  i  08 
.009  2041 
.009  2974 
.009  3906 

y  53- 
933 
933 
932 
932 

.0156 

!o!58 
.0159 

.014  6546 
.014  7460 
.014  8374 
.014  9288 

914 
914 
914 
914 

0.0040 

0.003  8332 

A  ?  1 

O.OIOO 

0.009  4838 

C\  1  1 

0.0160 

0.015  °202 

.0041 

.003  9284 

952 

.0101 

.009  5770 

y^-6 

.0161 

.015  1115 

9  3 

.0042 

.004  0235 

951 

.0102 

.009  6702 

932 

r\  *  T 

.0162 

.015  2028 

n  T  -9 

.0043 
.0044 

.004  i  i  86 
.004  2136 

950 
95° 

.0103 

.0104 

.009  7633 
.009  8564 

931 
931 
93I 

.0163 
.0164 

.015  2941 

•015  3854 

y1  3 
913 
912 

0.0045 
.0046 
.0047 
.0048 
.0049 

0.004  3086 
.004  4036 

.004  4985 

.004  5934 
.004  6883 

950 
949 
949 
949 
949 

0.0105 
.0106 
.0107 
.0108 
.0109 

0.009  9495 
.010  0425 
.010  1355 
.010  2285 
.010  3215 

93° 
93° 
93° 
93° 
929 

0.0165 
.0166 
.0167 
.0168 
.0169 

0.015  4766 
.015  5678 
.015  6589 
.015  7500 
.015  8411 

912 
911 
911 
911 
911 

0.0050 
.0051 
.0052 
.0053 
.0054 

0.004  7832 
.004  8780 
.004  9728 
.005  0675 
.005  1622 

948 

948 
947 
947 
947 

O.OIIO 
.0111 
.0112 
.0113 
.0114 

o.oio  4144 
.010  5073 
.010  6001 
.010  6929 
.010  7857 

929 

928 
928 
928 
928 

0.0170 
.0171 
.0172 
.0173 
.0174 

0.015  9322 
.016  0232 
.016  1142 
.016  2052 
.016  2961 

910 
910 
910 

909 
909 

0.0055 
.0056 
.0057 
.0058 
.0059 

0.005  ^569 
•005  3515 

.005  4461 
.005  5407 
.005  6353 

946 
946 
946 
946 
945 

O.OII5 
.0116 
.0117 
.0118 
.OII9 

o.oio  8785 
.010  9712 
.on  0639 
.on  1565 
.on  2491 

927 
927 
926 
926 
926 

0.0175 
.0176 
.0177 
.0178 

.0179 

0.016  3870 
.016  4779 
.016  5688 
.016  6596 
.016  7504 

909 
909 
908 
908 
908 

O.0060 

0.005  7298 

0.0120 

o.on  3417 

0.0180 

0.016  8412 

624 


TABLE  XIII 

For  finding  the  Eatio  of  the  Sector  to  the  Triangle. 


n 

logs* 

Diff. 

1? 

log  si 

Diff. 

>? 

logs* 

Diff. 

0.0180 
.0181 
.0182 

0.016  8412 
.016  9319 
.017  0226 

907 

907 

0.0240 
.0241 
.0242 

0.022  2330 
.022  3220 
.022  4109 

890 
889 

oo' 

0.0300 
.0301 
.0302 

0.027  52*8 
.027  6091 
.027  6964 

873 
873 

.0183 
.0184 

.017  1133 
.017  2039 

907 
906 
906 

.0243 
.0244 

.022  4998 
.022  5887 

009 

889 
889 

.0303 
.0304 

.027  7836 
.027  8708 

872 
872 
872 

0.0185 
.0186 
.0187 

0.017  ^945 
.017  3851 
.017  4757 

906 
906 

0.0245 
.0246 
.0247 

0.022  6776 
.022  7664 
.022  8552 

888 

888 

ooo 

0.0305 
.0306 

.0307 

0.027  9580 
•  .028  0452 
.028  1323 

872 

I71 

.0188 
.0189 

.017  5662 
.017  6567 

9°5 
905 
904 

.0248 
.0249 

.022  9440 
.023  0328 

oofl 

888 
887 

.0308 

.0309 

.028  2194 
.028  3065 

871 
871 
871 

0.0190 
.0191 
.0192 
.0193 

.0194 

0.017  7471 
.017  8376 
.017  9280 
.018  0183 
.018  1087 

905 
904 
903 
904 
9°3 

O.O25O 
.0251 
.0252 
.0253 
.0254 

0.023  I2I5 

.023  2IO2 
.023  2988 
.023  3875 
.023  4761 

887 
886 
887 
886 
886 

0.0310 
.0311 

.0312 

.0313 
.0314 

0.028  3936 
.028  4806 
.028  5676 
.028  6546 
.028  7415 

870 
870 
870 
869 
869 

0.0195 
.0196 
.0197 
.0198 
.0199 

0.018  1990 
.018  2893 
.018  3796 
.018  4698 
.018  5600 

9°3 
903 
902 
902 
901 

0.0255 
.0256 
.0257 
.0258 
.0259 

0.023  5647 
.023  6532 
.023  7417 
.023  8302 
.023  9187 

885 

885 
885 
885 
884 

0.0315 

.0316 

.0317 
.0318 

.0319 

0.028  8284 
.028  9153 

.029  0022 
.029  0890 
.029  I758 

869 
869 
868 
868 
868 

0.0200 
.0201 
.0202 
.0203 
.0204 

0.018  6501 
.018  7403 
.018  8304 
.018  9205 
.019  0105 

902 
901 
901 
900 
900 

0.0260 
.026l 
.0262 
.0263 
.0264 

0.024  °°7I 
.024  0956 
.024  1839 
.024  2723 
.024  3606 

885 

883 
884 
883 
883 

0.0320 
.0321 
.0322 

.0323 
.0324 

0.029  2626 

.029  3494 
.029  4361 
.029  5228 
.029  6095 

868 
867 
867 
867 
866 

0.0205 
.0206 

0.019  1005 
.019  1905 

900 

0.0265 
.0266 

0.024  4489 
.024  5372 

883 

00^ 

0.0325 
.0326 

0.029  ^961 
.029  7827 

866 

O  /I  £ 

.0207 
.0208 
.0209 

.019  2805 
.019  3704 
.019  4603 

QOO 

899 
899 
899 

.0267 
.0268 
.0269 

.024  6254 
.024  7136 
.024  8018 

o  o  2 
882 
882 
882 

.0327 
.0328 
.0329 

.029  8693 
.029  9559 

.030  0424 

oDO 

866 
865 
866 

0.0210 
.0211 

0.019  5502 
.019  6401 

III 

0.0270 
.0271 

0.024  8900 
.024  9781 

881 

OO  f 

0.0330 

.0331 

0.030  1290 
.030  2154 

864 

Of.  \ 

.0212 

.019  7299 

oQo 

8n°, 

.0272 

.025  0662 

OO  1 

00, 

.0332 

.030  3019 

SOS 

Of. 

.0213 

.019  8197 

090 

807 

.0273 

.025  1543 

oo  1 

880 

•°333 

.030  3883 

a  04 
864 

.0214 

.019  9094 

°7  / 

898 

.0274 

.025  2423 

880 

•°334 

.030  4747 

864 

0.0215 
.0216 
.0217 
.0218 
.0219 

0.019  9992 
.020  0889 

.020  1785 
.020  2682 
.020  3578 

897 
896 
897 
896 
896 

0.0275 
.0276 
.0277 
.0278 
.0279 

0.025  33°3 
.025  4183 
.025  5063 
.025  5942 
.025  6821 

880 
880 

879 
879 
879 

0.0335 
•0336 

•0337 
.0338 

•°339 

0.030  5611 
.030  6475 
.030  7338 
.030  8201 
.030  9064 

861 
863 
863 
863 
862 

O.0220 

0.020  4474 

o_  « 

O.O28O 

0.025  7700 

9.Tn 

0.0340 

0.030  9926 

8fi-7 

.O22I 

.0222 
.0223 
.0224 

.020  5369 

.020  6264 

.020  7159 

.020  8054 

895 
895 
895 
895 
894 

.0281 
.0282 
.0283 
.0284 

.025  8579 
.025  9457 
.026  0335 
.026  1213 

079 

878 
878 
878 
877 

.0341 
.0342 

•°343 
.0344 

.031  0788 
.031  1650 
.031  2512 
•031  3373 

OO2 
862 
862 

861 

861 

O.0225 
.0226 
.0227 
.0228 
.0229 

0.020  8948 

.020  9842 

.021  0736 
.021  1630 
.O2I  2523 

894 

894 
894 
893 
893 

0.0285 
.0286 
.0287 
.0288 
.0289 

0.026  2090 
.026  2967 
.026  3844 
.026  4721 
.026  5597 

877 
877 

877 

876 
876 

0.0345 
.0346 

•°347 
.0348 
.0349 

0.031  4234 
.031  5095 

•031  5956 
.031  6816 
.031  7676 

861 

861 
860 
860 
860 

0.0230 
.0231 
.0232 
•0233 
.0234 

0.021  3416 
.O2I  4309 
.021  5201 
.O2I  6093 
.021  6985 

893 

892 
892 
892 
891 

0.0290 
.0291 
.0292 
.0293 
.0294 

0.026  6473 
.026  7349 
.026  8224 
.026  9099 
.026  9974 

876 

875 
875 
875 
875 

0.0350 
.0351 
.0352 

•°353 
.0354 

0.031  8536 
.031  9396 
.032  0255 
.032  1114 
.032  1973 

860 

859 
859 

Hi 

0.0235 
.0236 
.0237 
.0238 
.0239 

0.021  7876 
.021  8768 
.021  9659 
.022  0549 
.022  1440 

892 
891 
890 
891 
890 

0.0295 
.0296 
.0297 
.0298 
.0299 

0.027  0849 
.027  1723 
.027  2597 
.027  3471 
.027  4345 

874 
874 

874 
874 
873 

o-0355 
.0356 

.0357 
.0358 
.0359 

0.032  2831 
.032  3689 
.032  4547 
.032  5405 
.032  6262 

858 
858 
858 

857 
858 

0.0240 

0.022  2330 

0.0300 

0.027  5218 

0.0360!  0.032  7120 

40 


625 


TABLE  XIII, 

For  finding  the  Ratio  of  the  Sector  to  the  Triangle. 


i 

log«> 

Diff. 

» 

*, 

Diff. 

< 

log*» 

Diff. 

0.0360 
.0361 
.0362 
.0363 
.0364 

0.032  7120 
.032  7976 
.032  8833 
.032  9689 

.033  0546 

856 

857 
856 

857 
855 

0.060 
.061 
.062 
.063 
.064 

0.052  5626 
.053  3602 
.054  1556 
.054  9488 

•055  7397 

7976 
7954 
7932 
7909 
7888 

0.120 
.121 
.122 
.123 
.124 

0.096  8849 
.097  5692 
.098  2520 
.098  9331 
.099  6127 

6843 
6828 

6811 
6796 
6780 

mvO  t^-OO  ON 

CO  CO  CO  CO  CO 

O  O  O  O  O 
0  ' 

0.03^3  1401 

.033  2257 
.033  3112 
.033  3967 
.033  4822 

856 

855 
855 
855 
855 

0.065 
.066 
.067 
.068 
.069 

0.056  5285 
.057  3150 
.058  0994 
.058  8817 
.059  6618 

7865 
7844 
7823 
7801 
7780 

0.125 
.126 
.127 
.128 
.129 

o.ioo  2907 
.100  9672 
.101  6421 

.102  3154 
.102  9873 

6765  > 
6749 

6733 
6719  , 
6703 

0.0370 
.0371 
.0372 

•0373 
.0374 

0.033  5677 
.033  6531 
.033  7385 
.033  8239 
.033  9092 

854 
854 
854 
853 
854 

0.070 
.071 
.072 
.073 
.074 

0.060  4398 
.061  2157 
.061  9895 
.062  7612 
.063  5308 

7759 
7738 

7696 
7676 

0.130 
.131 
.132 

•'33 
•'34 

O.IO3  6576 
.104  3264 
.104  9936 
.105  6594 
.106  3237 

6688 
6672  ! 
6658  ! 
6643 
6628 

0.0375 
.0376 
.0377 
•0378 
•0379 

0.033  9946 
.034  0799 
.034  1651 
.034  2504 
.034  3356 

852 
853 
852 
852 

0.075 
.076 
.077 
.078 
•°79 

0.064  2984 
.065  0639 
.065  8274 
.066  5888 
.067  3482 

7655 
7635 
7614 

7594 
7575 

0.135 
.136 

•137 
.138 

•'39 

0.106  9865 
.107  6478 
.108  3076 
.108  9660 
.109  6229 

6613 
6598 
6584 
6569 
6554 

O  M  O  rorj- 
oo  oo  oo  oo  oo 

CO  CO  CO  CO  CO 

q  q  q  q  q 
d 

0.034  4208 

.034  5059 
.034  5911 

.034  6762 

.034  7613 

851 

852 
851 
851 
851 

0.080 
.081 
.082 
.083 
.084 

0.068  1057 
.068  8612 
.069  6146 
.070  3661 
.071  1157 

7555 

7534 

7496 
7476 

0.140 
.141 
.142 

•143 
.144 

o.no  2783 
.no  9323 

.III  5849 

.112  2360 
.112  8857 

6540 
6526 
6511 
6497 
6483 

vnvo  r--oo  ON 

00  00  00  00  00 
CO  CO  CO  CO  CO 

0  0  0  O  0 
0 

0.034  8464 

•°34  93H 
.035  0164 
.035  1014 
.035  1864 

850 
850 
850 
850 
849 

0.085 
.086 
.087 
.088 
.089 

0.071  8633 
.072  6090 

•073  3527 
.074  0945 

•°74  8345 

7457 
7437 
7418 
7400 
738o 

0.145 
.146 
.147 
.148 
.149 

0.113  534° 
.114  1809 
.114  8264 
.115  4704 
.116  1131 

6469 

6455 
6440 

6427 
6413 

0.0390 
.0391 

0.035  2713 
•035  3562 

849 

0.090 
.091 

0-075  5725 
.076  3087 

7362 

0.150 
.151 

0.116  7544 
•"7  3943 

6399| 

.0392 
.0393 
•0394 

•035  44" 
•035  5259 
.035  6108 

848 

849 
848 

.092 

•°93 
.094 

.077  0430 
.077  7754 
.078  5060 

7343 
7324 
7306 
7288 

.152 
•153 
•'54 

.118  0329 
.118  6701 
.119  3059 

6372 
6358 
6345  i 

0.0395 
.0396 

•°397 
.0398 

•°399 

0.035  6956 
.035  7804 
.035  8651 
.035  9499 
.036  0346 

848 

847 
848 

847 
846 

0.095 
.096 
.097 
.098 
.099 

0.079  2348 
.079  9617 
.080  6868 
.081  4101 
.082  1316 

7269 
7251 

7233 
7215 
7197 

0.155 
.156 

•'57 
.158 

•'59 

0.119  94°4 
•120  5735 

.121  2053 
.121  8357 
.122  4649 

6331 
6318 
6304  | 
6292 
6278  | 

0.040 
.041 
.042 
•°43 
•°44 

0.036  1192 
.036  9646 
•037  8075 
.038  6478 
.039  4856 

8454 
8429 
8403 
8378 
8353 

O.I  00 
.101 
.102 
.103 
.104 

0.082  8513 
.083  5693 
.084  2854 
.084  9999 
.085  7125 

7180 
7161 

7H5 
7126 
7110 

0.160 
.161 
.162 
.163 
.164 

0.123  0927 
.123  7192 

.124  3444 
.124  9682 
.125  5908 

6265 
6252 
6238 
6226 
6213 

0.045 
.046 
.047 
.048 
•049 

0.040  3209 
.041  1537 
.041  9841 
.042  8121 
.043  6376 

8328 
8304 
8280 
8255 
8231 

0.105 
.IO6 
.107 
.108 
.109 

0.086  4235 
.087  1327 
.087  8401 
.088  5459 
.089  2500 

7092 
7074 
7058 
7041 
7023 

0.165 
.166 
.167 
.168 
.169 

0.126  ai2i 
.126  8321 
.127  4508 
.128  0683 
.128  6845 

6200  ; 
6187  ! 
6175 
6162 
6149 

0.050 
.051 
.052 
.053 
.054 

0.044  46°7 
.045  2814 
.046  0997 
.046  9157 
.047  7294 

8207 
8183 
8160 

8113 

O.IIO 

.III 

.112 

•"3 

.114 

0.089  9523 
.090  6530 
.091  3520 
.092  0494 
.092  7451 

7007 
6990 
6974 
6957 
6940 

0.170 

.171 

.172 

.173 
.174 

0.129  2994 
.129  9131 

•13°  5255 
.131  1367 
.131  7466 

6137 
6124 
6112 
6099 
6087 

0.055 

.056 
.057 
.058 
.059 

0.048  5407 

•049  3496 
.050  1563 
.050  9607 
.051  7628 

8089 
8067 
8044 
8021 
7998 

0.115 
.116 
.117 
.118 

.119 

0.093  4391 
.094  1315 
.094  8223 
.095  5114 
.096  1990 

6924 
6908 
6891 
6876 
6859 

0.175 
.176 
.177 
.178 
.179 

0-132  3553 
.132  9628 
.133  5690 
.134  1740 
•134  7778 

6062 
6050 
6038 
6026 

O.O60 

0.052  5626 

0.120 

0.096  8849 

0.180 

0.135  3804 

626 


TABLE  XIII. 

For  finding  the  Ratio  of  the  Sector  to  the  Triangle. 


* 

logs* 

Diff. 

* 

logs* 

Diff. 

i 

toff* 

Diff. 

o.i8o 
.181 
.182 
.183 
.184 

0.135  3804 

.135  9818 

.136  5821 
.137  1811 
.137  7789 

6014 
6003 
5990 
5978 
5966 

O.240 
.241 
.242 

•*43 

.244 

0.169  509* 
.170  0470 

.170  5838 
.171  1197 
.171  6547 

5378 
5368 

5359 
5350 
534° 

0.300 
.301 
.302 
.303 
•3°4 

0.200  2285 
.200  7157 
.201  2021 
.201  6878 
.202  1727 

4872 
4864 
4857 
4849 
4842 

0.185 
.186 
.187 
.188 
.189 

0-138  3755 
.138  9710 

•!39  5653 
.140  1585 
.140  7504 

5955 
5943 
593* 
5019 
5908 

0.245 
.246 
.247 
.248 
.249 

0.172  1887 
.172  7218 

.173  2540 

.173  7853 
.174  3156 

15322 

|53'3 
15303 
5*95 

0.305 
.306 

•3°7 
.308 
.309 

O.2O2  6569 
.203  1403 
.203  6230 
.204  1050 
.204  5862 

4834 
4827 
4820 
4812 
4805 

0.190 
.191 

0.141  3412 
.141  9309 

5897 
c88  c 

0.250 
.251 

0.174  8451 

.175  3736 

5285 

O.3IO 
.311 

0.205  0667 
.205  5464 

4797 

.192 

•193 
.194 

.142  5194 
.143  1068 
.143  6931 

5874 
5863 
5851 

.252 

•*53 
.254 

.175  9013 

.176  4280 
.176  9538 

5*77 
5267 
5258 
5*5° 

.312 

•3'3 
•3'4 

.206  0254 
.206  5037 
.206  9813 

4790 
4783 
4776 
4768 

0.195 
.196 
.197 
.198 
.199 

0.144  2782 
.144  8622 
.145  4450 
.146  0268 
.146  6074 

5840 
5828 
5818 
5806 
5795 

0.255 
.256 
.257 
.258 
.259 

0.177  4788 

.178  0029 

.178  5261 
.179  0484 
.179  5698 

5*3* 
5**3 
5214 

5*°5 

0-3'5 
.316 

:\\l 

•3*9 

O.207  4581 
.207  9342 
.208  4096 
.208  8843 
.209  3582 

4761 
4754 
4747 
4739 
4733 

O.200 
.201 

0.147  *86g 
.147  7653 

5784 

0.260 
.261 

0.180  0903 
.180  6100 

5*97 

0.320 
•3*i 

0.209  8315 
.210  3040 

47*5 

.202 
.203 
.204 

.148  3427 
.148  9189 
.149  4940 

5774 
5762 

5751 
574* 

.262 
.263 
.264 

.181  1288 

.181  6467 

.182  1638 

5188 

5*79 
5171 
5162 

.322 
•3*3 
•3*4 

.210  7759 

.211  2470 
.211  7174 

4719 
4711 
4704 
4697 

O.20C 

0.150  0681 

0.265 

0.182  6800 

0.325 

O.2I2  1871 

.206 

.150  6411 

5730 

.266 

.183  1953 

5153 

.212  6562 

4691 

.207 
.208 
.209 

.151  2130 
.151  7838 
•'5*  3535 

57X9 
5708 
5697 
5687 

.267 
.268 
.269 

.183  7098 
.184  2235 
.184  7363 

5*45 
5'37 
5128 
5120 

•3*7 
•3*8 
•3*9 

.213  1245 
.213  5921 
.214  0591 

4676 
4670 
4662 

0.210 
.211 
.212 
.213 

.214 

0.152  9222 
.153  4899 
.154  0565 
.154  6210 
.155  1865 

5677 
5666 
5655 
5645 
5634 

0.270 
.271 
.272 

•*73 
.274 

0.185  2483 
.185  7594 
.186  2696 
.186  7791 
.187  2877 

5111 

5102 

5086 
5078 

0.330 
•33* 
•33* 
•333 
•334 

0*214  5*53 
.214  9909 
.215  4558 
.215  9200 
.216  3835 

4656 
4649 
4642 

4635 
4629 

O.2I5 
.216 
.217 
.218 
.219 

o-i55  7499 
.156  3123 
.156  8737 
.157  4340 
•'57  9933 

5624 
5614 
5603 
5593 
5583 

0.275 
.276 
.277 
.278 
.279 

0.187  7955 
.188  3024 
.188  8085 
.189  1138 
.189  8183 

5069 
5061 
5°53 
5°45 
5°37 

0-335 
•336 
•337 
•338 
•339 

0.216  8464 
.217  3085 
.217  7700 
.218  2308 
.218  6910 

4621 
4615 
4608 
4602 

0.220 
.221 
.222 
.223 
.224 

0.158  5516 
.159  1089 
.159  6652 
.160  2204 
.160  7747 

5573 
5563 
555* 
5543 
553* 

0.280 
.281 
.282 
.283 
.284 

0.190  3220 
.190  8249 
.191  3269 
.191  8281 
.192  3286 

5029 

5020 
5012 
5005 
4996 

0.340 
•341 
•34* 
•343 
•344 

0.219  1505 
.219  6093 
.220  0675 

.220  5250 
.220  9818 

4588 
4582 

4575 
4568 
4562 

0.225 
.220 
.227 
.228 
.229 

0.161  3279 
.161  8802 
.162  4315 
.162  9817 
.163  5310 

55*3 
5513 
5502 

5493 
5483 

0.285 
.286 
.287 
.288 
.289 

0.192  8282 
.193  7271 
.193  8251 
.194  3224 
.194  8188 

4989 
4980 

4973 
4964 

4957 

0-345 
.346 

•347 
.348 

•349 

0.221  4380 
.221  8915 
.222  7483 
.222  8025 
,223  2561 

4555 
4548 
4542 
4536 

0.230 
.231 

.232 

•*33 
•*34 

0.164  0793 
.164  6267 
.165  1710 
.165  7184 
.166  2628 

5474 
5463 

5454 
5444 
5435 

0.290 
.291 
.292 
.293 
.294 

0.195  3145 
.195  8094 
.196  3035 
.196  7968 
.197  2894 

4949 

494  1 

4933 
(.926 

0.350 
•35' 
•35* 
•353 
•354 

0.223  7090 
.224  1613 
.224  6130 
.225  0640 
.225  5143 

45*3 
45^7 
4510 

45°3 
4497 

0.235 
.236 

0.166  8063 
.167  3488 

54*5 

0.295 
.296 

0.197  7811 
.198  2721 

1.910 

0-355 
•356 

0.225  964° 
.226  4131 

449  > 
4484 

•*37 

.167  8903 

r^nfi 

.297 

.198  7624 

I-903 

A9nA 

•357 

.226  8615 

.238 

.168  4309 

400 

.298 

.199  2518 

1.094 
l888 

•358 

.227  3093 

AA72 

.239 

.168  9705 

5396 
387 

.299 

.199  7406 

4879 

•359 

.227  7565 

4--j-  /  * 

4466 

0.240 

0.169  5°9* 

0.300 

0.200  2285 

0.360 

0.228  2031 

627 


TABLE  XIII, 

For  finding  the  Katio  of  the  Sector  to  the  Triangle. 


* 

logss 

Diff. 

< 

log  «" 

Diff. 

* 

logs* 

Diff. 

0.360 
.36i 
.362 
.363 
•364 

0.228  2031 
.228  6490 

.229  0943 
.229  5390 
.229  9831 

4459 
4453 
4447 
4441 

4434 

0.420 
.421 
.422 
•4*3 
•4*4 

°-*53  9T53 
.254  3269 

•*54  7379 
.255  1484 

•*55  5584 

4116 

4110 
4105 
4100 
4095 

0.480 
.48i 
.482 
.483 
.484 

0.277  7*7* 
.278  1096 
.278  4916 
.278  8732 
•*79  *543 

3824 
3820  i 
3816 
3811 
3806 

U^SO  t^OO  O 
SO  so  SO  so  VO 
CO  CO  CO  CO  CO 

6  * 

0.230  4265 
.230  8694 
.231  3116 

.231  7532 

.232  1942 

4429 
4422 
4416 
4410 
44°4 

0.425 
.426 

•4*7 
.428 

•4*9 

0.255  9679 
.256  3769 
.256  7853 
.257  1932 
.257  6006 

4090 
4084 
4079 
4074 
4069 

0.485 
.486 
.487 
.488 
•489 

0.279  6349 
.280  0151 
.280  3949 
.280  7743 
.281  1532 

3802 
3798 
3794 
3785 
3784 

0.370 
.371 

•37* 

0.232  6346 
•*33  °743 
•*33  5135 

4397 
439* 

Atxn 

0430 
•431 
•43* 

0.258  0075 
.258  4139 
.258  8198 

4064 
4059 

0.490 
.491 

.492 

0.281  5316 
.281  9096 
.282  2872 

378o 
3776 

•373 
•374 

.233  9521 
.234  3900 

4300 

4379 
4374 

•433 
•434 

.259  2252 
.259  6300 

4054 
4048 
4044 

•493 
•494 

.282  6644 
.283  0411 

377* 
3767 
3762 

0.375 
•376 
•377 
.378 

•379 

0.234  8274 
.235  2642 
.235  7003 
.236  1359 
.236  5709 

4368 
4361 
4356 
435° 

0.435 
•436 
•437 
•438 
•439 

0.260  0344 
.260  4382 
.260  8415 
.261  2444 
.261  6467 

4038 

4033 
4029 
4023 

o-495 
•496 
•497 
.498 

•499 

0.283  4173 
.283  7932 
.284  1686 
.284  5436 
.284  9181 

3759 

3754  ! 
375°  i 
3745 

4344 

4019 

374* 

O  M  M  COrJ- 
OO  00  00  00  00 
CO  tO  CO  CO  CO 

o 

0.237  0053 
.237  4391 
.237  8723 

.238  3050 

.238  7370 

4338 
433* 

4327 
4320 

0.440 
•44  1 
•44* 
•443 
•444 

0.262  0486 
.262  4499 
.262  8507 
.263  2511 
.263  6509 

4013 
4008 
4004 
3998 

0.500 
.501 
.502 
.503 
.504 

0.285  *9*3 
.285  6660 
.286  0392 
.286  4121 
.286  7845 

3737 
373*  I 
37^9 

37^4  . 

43*  5 

3994 

3720 

tJ->sO  t--OO  O 

oo  oo  oo  oo  oo 

CO  CO  CO  tO  tO 

6  * 

0.239  *685 

•*39  5993 
.240  0296 
.240  4594 
.240  8885 

4308 

43°3 
4298 
4291 
4286 

0.445 
•446 
•447 
•448 
•449 

0.264  °5°3 
.264  4492 
.264  8475 
.265  2454 
.265  6428 

3989 
3983 
3979 
3974 
3969 

0.505 
.506 

•507 
.508 
.509 

0.287  ^65 
.287  5281 
.287  8992 
.288  2700 
.288  6403 

3716 

37"  ; 

3708  i 

3703 
3699  1 

0.390 
•391 

•39* 
•393 
•394 

0.241  3171 
.241  7451 
.242  1725 
.242  5994 
.243  0257 

4280 

4*74 
4269 

4*63 
4*57 

0.450 
•45  * 
•45* 
•453 
•454 

0.266  0397 
.266  4362 
.266  8321 
.267  2276 
.267  6226 

3965 
3959 
3955 
395° 
3945 

0.510 
.511 

.512 

.514 

0.289  OIO2 

.289  3797 
.289  7487 
.290  1174 
.290  4856 

3695 
3690  ; 
3687  i 
3682 
3679 

0.395 
•396 

•397 

•398 

•399 

0.243  45H 
.243  8766 
.244  3012 
.244  7252 
.245  1487 

4*5* 
4246 

4240 

4*35 
4229 

0.455 
•456 
•457 
•458 
•459 

0.268  0171 
.268  4111 
.268  8046 
.269  1977 
.269  5903 

3940 
3935 
3931 
3926 
3921 

0.515 
.516 

•517 
.518 
.519 

0.290  8535 
.291  2209 
.291  5879 
.291  9545 
.292  3207 

3674 
3670 
3666 

3662  ; 
3657 

0.400 

.401 

.402 

•403 

0.245  57J6 
.245  9940 
.246  4158 
.246  8371 

4224 
4218 

0.460 
.461 
.462 
•463 

0.269  9824 
.270  3741 
.270  7652 
.271  1559 

39*7 
39" 

3907 

0.520 
.521 
.522 
•5*3 

0.292  6864 
.293  0518 
.293  4168 
.293  7813 

3654 

365°  : 
3645 

•404 

.247  2578 

4207 
4201 

.464 

.271  5462 

3898 

•5*4 

.294  1455 

3642 
3637 

0.405 
.406 
.407 

.408 
•409 

0.247  6779 
.248  0975 
.248  5166 
.248  9351 
.249  3531 

4196 
4191 

4*85 
4180 

4*74 

0.465 
.466 

•467 
.468 
.469 

0.271  9360 
.272  3253 
.272  7141 
.273  1025 
.273  4904 

mi 
3884 
3879 
3874 

0-5*5 
.526 

•5*7 
.528 

•5*9 

0.294  5°9* 
.294  8726 
.295  2355 
.295  5981 
.295  9602 

3634 
36291 
3626 

3621  ; 
3618 

0.410 

.411 
.412 
.413 
.414 

0.249  77°5 
.250  1874 
.250  6038 
.251  0196 
.251  4349 

4169 
4164 
4158 
4*53 
4*47 

0.470 
.471 

•47* 
•473 
•474 

0.273  8778 
.274  2648 
.274  6513 

•*75  °374 
.275  4230 

3870 
3865 
3861 
3856 
385* 

0-53° 
•531 
•53* 
•533 
•534 

0.296  3220 
.296  6833 
.297  0443 

•*97  4°49 
.297  7650 

3613 
3610  i 
3606 
3601 
3598 

0.415 

.416 

.417 
.418 
.419 

0.251  8496 
.252  2638 
.252  6775 
.253  0906 
.253  5032 

4142 

4*37 
4131 
4126 
4121 

0.475 
•476 
•477 
.478 
•479 

0.275  8082 
.276  1929 
.276  5771 
.276  9609 
.277  3443 

3847 
3842 
3838 

3834 
3829 

0-535 
•536 
•537 
.538 
•539 

0.298  1248 
.298  4842 
.298  8432 
.299  2018 
.299  5600 

3594 
359° 
3586 
358* 
3578 

0.420 

0.253  9*53 

0.480 

0.277  7*7* 

0.540 

0.299  9*78 

628 


TABLE  XIIL 

For  finding  the  Katio  of  the  Sector  to  the  Triangle. 


1? 

logs* 

Diff. 

* 

log*a 

Diff. 

* 

log«» 

Diff. 

0.540 

•54» 
•54* 
•543 

0.199  9178 
.300  2752 
.300  6323 
.300  9890 

3574 
3567 

0.560 
.561 
.562 
•563 

0.306  9938 

•3°7  3437 
.307  6931 
.308  0422 

3499 
3494 

0.580 
.581 
.582 
•583 

0.313  9215 
.314  26^.1 
.314  6064 

.314  9481 

34*6 
34*3 
3419 

•544 

.301  3452 

3559 

.564 

.308  3910 

*484 

•584 

.315  2898 

3412 

0-545 
b$4* 

•547 
.548 
•549 

0.301  7011 
.302  0566 
.302  4117 
.302  7664 
.303  1208 

3555 
3551 
3547 
3544 
3540 

0.565 
.566 
.567 
.568 
.569 

0.308  7394 
.309  0874 
.309  4150 
.309  7823 
.310  1292 

3480 
3476 

3473 
3469 

3466 

0.585 
.586 
.587 
.588 
.589 

0.315  6310 

•3'5  9719 
.316  1124 
.316  6525 
.316  9923 

3409 
3405 
3401 

3398 
3395 

0.550 
•551 
•55* 
•553 
•554 

0-303  4748 
.303  8284 
.304  1816 
.304  5344 
.304  8869 

3536 
353* 
35*8 
35*5 

0.570 
•571 
•57* 
•573 
•574 

0.310  4758 
.310  8220 
.311  1678 
•311  5i33 

3462 
3458 
3455 

345  i 
3447 

0.590 
.591 
•59* 
•593 
•594 

0.317  3318 
.317  6709 
.318  0096 
.318  3480 
.318  6861 

3387 

3384 
3381 

3377 

0-555 

•556 
•557 
•558 
•559 

0.305  2390 
.305  5907 
.305  9420 
.306  2930 
.306  6436 

35'7 
35*3 
3510 
3506 
3502 

0-575 
•576 
•577 
.578 
•579 

0.312  2031 

•31*  5475 
.312  8915 
.313  2352 
•3'3  5785 

3444 
3440 

3437 
3433 
343° 

0.595 
•596 
•597 
•598 
•599 

0.319  0238 
.319  3612 
.319  6983 
.320  0350 
•3*o  37H 

3374 

3367 
3364 
3360 

0.560 

0.306  9938 

0.580 

0.313  9215 

0.600 

0.320  7074 

TABLE  XIV. 

For  finding  the  Ratio  of  the  Sector  to  the  Triangle. 


X 

t 

X 

f 

Ellipse. 

Diff. 

Hyperbola. 

Diff. 

Ellipse. 

Diff. 

Hyperbola. 

Diff. 

o.ooo 

0.000  0000 

o.ooo  oooo 

I 

0.030 

o.ooo  0523 

76 

o.ooo  0506 

11 

.001 

.000  0001 

1 

.000  0001 

J 

.031 

.000  0559 

3 

.000  0539 

76 

.002 

.OOO  OOO2 

.OOO  OOO2 

.032 

.000  0596 

8 

.000  0575 

11 

.003 

.000  0005 

3 

.000  0005 

3 

•033 

.000  0634 

3 

.000  06  1  1 

3U 

MM 

.004 

.000  0009 

4 
5 

.000  0009 

4 
5 

.034 

.000  0674 

4° 

40 

.000  0648 

3? 

O.005 

o.ooo  0014 

o.ooo  0014 

0.035 

o.ooo  0714 

o.ooo  0686 

.006 

.OOO  OO2I 

7 

.OOO  OO2O 

.036 

.000  0756 

42 

.000  0726 

40 

.007 

.000  0028 

7 

.000  0028 

•037 

.000  0799 

43 

.000  0766 

4° 

.008 
.009 

.000  0037 
.000  0047 

9 

10 

II 

.000  0036 
.000  0046 

10 

II 

.038 
•039 

.000  0844 
.000  0889 

45 
45 
47 

.000  0807 
.000  0850 

41 
43 
44 

O.OIO 
.Oil 

o.ooo  0058 
.000  0070 

12 

o.ooo  0057 
.000  0069 

12 

0.040 
.041 

o.ooo  0936 
.000  0984 

48 

o.ooo  0894 
.000  0978 

.012 

.013 

.000  0083 
.000  0097 

i! 

.000  0082 
.000  0096 

*3 

14 

.042 
.043 

.000  1033 
.000  1084 

49 

51 

.000  0984 
.000  1031 

$ 

i  .014 

.000  0113 

ID 

17 

.OOO  01  I  I 

11 

•044 

.000  1135 

53 

.000  1079 

49 

0.015 

!   .Ol6 

o.ooo  0130 
.000  0148 

18 

o.ooo  0127 
.000  0145 

IJB 

0.045 

.046 

o.ooo  1  1  88 
.000  1242 

It 

o.ooo  1128 
.000  1178 

5° 

;  .017 
.018 

.019 

.000  0167 
.000  0187 
.000  0209 

19 

20 
22 
22 

.000  0164 
.000  0183 
.000  0204 

19 
19 

21 

22 

.047 
.048 
.049 

.000  1298 
.000  1354 
.000  1412 

56 
56 

58 

59 

.000  1229 
.000  1281 
.000  1334 

51 

5* 
53 
55 

0.020 
.021 
.022 
.023 
.024 

o.ooo  0231 
.000  0255 
.000  0280 
.000  0306 
.000  0334 

24 

\\ 

28 
28 

o.ooo  0226 
.000  0249 
.000  0273 
.000  0298 
.000  0325 

23 
24 

27 

27 

0.050 

.051 
.052 

•053 
.054 

o.ooo  1471 
.000  1532 
.000  1593 
.000  1656 
.000  1720 

OS  ON  ON  ON  ON 
in  4*  OJ  HI  M 

o.ooo  1389 
.000  1444 
.000  1500 
.000  1558 
.000  1616 

VNVOOOOO  ON 
w>  10  u->  1/1  iri 

0.025 
.026 
.027 
.028 
.029 

o.ooo  0362 
.000  0392 
.000  0423 

.000  0455 
.000  0489 

3° 

31 

32 

34 
34 

o.ooo  0352 
.000  0381 
.000  0410 
.000  0441 
.000  0473 

29 

*9 

31 

3* 
33 

0.055 

.056 

.057 
.058 
•059 

o.ooo  1785 
.000  1852 
.000  1920 
.000  1989 
.000  2060 

67 

68 
69 

o.ooo  1675 
.000  1736 
.000  1798 

.000  i860 

.000  1924 

61 
62 
62 

64 
64 

0.030 

o.ooo  0523 

o.ooo  0506 

0.060  |  o.ooo  2131 

o.ooo  1988 

629 


TABLE  XIV. 

For  finding  the  Katio  of  the  Sector  to  the  Triangle. 


X 

* 

X 

e 

Ellipse. 

Diff. 

Hyperbola. 

Diff. 

Ellipse. 

Diff. 

Hyperbola. 

Diff. 

o.o6o 
.061 
.062 

o.ooo  2131 
.000  2204 
.000  2278 

73 

a 

o.ooo  1988 
.000  2054 

.000  2  1  21 

66 

67 

£0 

0.120 
.121 
.122 

o.ooo  8845 
.000  8999 
.000  9154 

154 
155 

o.ooo  7698 
.000  7822 
.000  7948 

124 
126 

_   f 

.063 
.064 

.000  2354 
.000  2431 

70 

.OOO  2189 

.000  2257 

Do 

68 

70 

.123 

.124 

.000  9311 
.000  9469 

J57 

158 

'59 

.000  8074 
.000  8202 

120 

128 
128 

0.065 
.066 
.067 
.068 
.069 

o.ooo  2509 
.000  2588 
.000  2669 
.000  2751 
.000  2834 

79 
81 
82 
83 
84 

o.ooo  2327 
.000  2398 
.oco  2470 
.000  2543 
.000  2617 

71 
72 

73 
74 
74 

0.125 
.126 
.127 
.128 
.129 

o.ooo  9628 
.000  9789 
.000  9951 
.001  0115 
.001  0280 

161 

162 

164 

'65 
167 

o.ooo  8330 
.000  8459 
.000  8590 
.000  8721 
.000  8853 

129 
131 

132 

133 

0.070 
.071 
.072 

o.ooo  2918 
,000  3004 
.000  3091 

86 
87 

o.ooo  2691 
.000  2767 
.000  2844 

76 
778 

0.130 
.131 
.132 

o.ooi  0447 
.001  0615 
.001  0784 

168 

169 

o.ooo  8986 
.000  9120 
.000  9255 

134 

.073 
.074 

.000  3180 
.000  3269 

89 
91 

.000  2922 
.000  3001 

1° 
79 
80 

•J33 
.134 

.001  0955 
.001  1128 

171 

173 

J73 

.000  9390 
.000  9527 

*35 

137 

0.075 
.076 

o.ooo  3360 
.000  3453 

93 

o.ooo  3081 
.000  3162 

Si 
82 

.136 

o.ooi  1301 
.001  1477 

176 

o.ooo  9665 
.000  9803 

138 

•077 

.000  3546 

93 

.000  3244 

•137 

.001  1654 

III 

.000  9943 

140 

.078 

.000  3641 

95 

.000  3327 

0^ 

.138 

.001  1832 

178 

IJ?0 

.001  0083 

140 

.079 

.000  3738 

97 
97 

.000  3411 

85 

.139 

.OOI  2OI2 

i  oo 
181 

.001  0224 

141 

142 

0.080 
.081 

o.ooo  3835 
.000  3934 

99 

o.ooo  3496 
.000  3582 

86 

8- 

0.140 
.141 

o.ooi  2193 
.001  2376 

183 

o.ooi  0366 
.001  0509 

143 

.082 

.000  4034 

IOO 

.000  3669 

y 

88 

.142 

.001  2560 

184 

.001.  0653 

144 

.083 
.084 

.000  4136 
.000  4239 

I  02 

103 

104 

.000  3757 
.000  3846 

89 
90 

.143 
.144 

.001  2745 
.001  2933 

188 
188 

.001  0798 
.001  0944 

'45 

146 
147 

0.085 
.086 
.087 
.088 

o.ooo  4341 
.000  4448 
.000  4555 
.000  4663 

105 
107 

108 

o.ooo  3936 
.000  4027 
.000  4119 
.000  4212 

91 
92 

93 

0.145 
.146 

.147 
.148 

o.ooi  3121 
.001  3311 

.001  3503 
.001  3696 

190 
192 
193 

o.ooi  1091 
.001  1238 
.001  1387 
.001  1536 

147 
149 
149 

.089 

.000  4773 

no 
in 

.000  4306 

94 
95 

.149 

.001  3891 

195 

196 

.001  1686 

150 

152 

0.090 
.091 

o.ooo  4884 
.000  4996 

112 

o.ooo  4401 
.000  4496 

95 

0.150 
.151 

o.ooi  4087 
.001  4285 

198 

o.ooi  1838 
.001  1990 

152 

.092 

•°93 
.094 

.000  5109 
.000  5224 
.000  5341 

"5 
117 

117 

.000  4593 
.000  4691 
.000  4790 

97 
98 
99 

IOO 

.152 
.154 

.001  4484 
.001  4684 

.001  4886 

199 

2OO 
2  O2 

204 

.001  2143 
.001  2296 
.001  2451 

*53 

153 

155 

156 

0.095 
.096 
.097 
.098 
.099 

o.ooo  5458 
.000  5577 
.000  5697 
.000  5819 
.000  5942 

119 

120 

122 
I23 
124 

o.ooo  4890 
.000  4991 
.000  5092 
.000  5195 
.000  5299 

101 
101 

103 
104 
104 

0.155 
.156 

•J57 
.158 
.159 

o.ooi  5090 
.001  5295 
.001  5502 
.001  5710 
.001  5920 

205 
207 
208 

2IO 
211 

o.ooi  2607 
.001  2763 
.001  2921 
.001  3079 
.001  3238 

156 

158 
158 

III 

O.IOO 
.101 

.102 
.103 
.104 

o.ooo  6066 
.000  6192 
.000  6319 
.000  6448 
.000  6578 

126 
I27 
I29 
130 
131 

o.ooo  5403 
.000  5509 
.000  5616 
.000  5723 
.000  5832 

1  06 

107 
107 

109 
109 

o.i  60 
.161 
.162 
.163 
.164 

o.ooi  6131 
.001  6344 
.001  6559 
.001  6775 
.001  6992 

213 
215 

216 

217 
219 

o.ooi  3398 
.001  3559 

.001  3721 
.001  3883 
.001  4047 

161 

162 
162 

164 

164 

O.I05 
.106 

.107 

,108 

o.ooo  6709 
.000  6842 
.000  6976 
.000  7111 

133 
134 
135 

o.ooo  5941 
.000  6052 
.000  6163 
.000  0275 

III 
III 

112 

0.165 
.166 
.167 
.168 

o.ooi  7211 
.001  7432 
.001  7654 
.001  7878 

221 
222 

224 

o.ooi  4211 
.001  4377 
.001  4543 
.001  4710 

166  ' 
166 

167 

_  fa 

.109 

.000  7248 

137 
138 

.000  6389 

114 

.169 

.001  8103 

225 

227 

.001  4878 

IOO 

169 

O.I  10 

.III 

o.ooo  7386 
.000  7526 

140 

o.ooo  6503 

.000  66l8 

115 

0.170 
.171 

o.ooi  8330 
.001  8558 

228 

o.ooi  5047 
.001  5216 

i£9 

.IJ2 

•"3 

.114 

.000  7667 
.000  7809 
.000  7953 

141 
I42 
144 
•45 

.000  6734 
.000  6851 
.000  6969 

1  1  6 
117 
118 
119 

.172 
.173 
.174 

.001  8788 
.001  9020 
.001  9253 

230 
232 

233 
234 

.001  5387 
.001  5558 
.001  5730 

'  /  * 

171 

172 
173 

0.115 

.116 

o.ooo  8098 
.000  8245 

147 
_  .  6 

o.ooo  7088 
.000  7208 

120 

0.175 
.176 

o.ooi  9487 
.001  9724 

237 

o.ooi  5903 
.001  6077 

*74 

.117 
.118 
.119 

.000  8393 
.000  8542 
.000  8693 

148 
149 

152 

.000  7329 
.000  7451 
.000  7574 

121 

122 

123 

124 

.177 
.178 
.179 

.001  9961 

.002  0201 

.002  0442 

237 

240 
241 
343 

.001  6252 
.001  6428 
.001  6604 

176 
176 
178 

0.120 

o.ooo  8845 

o.ooo  7698 

o.i  80 

0.002  0635 

o.ooi  6782 

630 


TABLE  XIY. 

For  finding  the  Katio  of  the  Sector  to  the  Triangle. 


X 

S 

X 

£ 

Ellipse. 

Diff. 

Hyperbola. 

Diff. 

Ellipse. 

Diff. 

Hyperbola. 

Diff. 

0.180 
.181 
.182 

.183 
.184 

0.002  0685 
.002  0929 

.002  1175 
.002  I42Z 
.002  1671 

244 
246 
247 
249 

"7  C  T 

o.ooi  6782 
.001  6960 
.001  7139 
.001  7319 
.001  7500 

178 
179 

180 
181 
181 

0.240 

.241 
.242 

•*43 
.244 

0.003  8289 
.003  8635 
.003  8983 
.003  9333 
.003  9685 

346 
348 
35° 
35* 

0.002  8939 
.002  9166 
.002  9394 
.002  9623 
.002  9852 

227 
228 
229 
229 

451 

354 

231 

0.185 
.186 
.187 
.188 
.189 

O.002  1922 
.OO2  2174 
.002  2428 
.OO2  2683 
.002  2941 

252 
254 

255 
258 
258 

o.ooi  7681 
.001  7864 
.001  8047 
.001  8231 
.001  8416 

183 
183 
184 
185 
186 

0.245 
.246 
.247 
.248 
.249 

0.004  °°39 
.004  0394 
.004  0752 
.004  ii  ii 
.004  1472 

355 
358 
359 
361 
363 

0.003  °°83 
.003  0314 
.003  0545 
.003  0778 
.003  ion 

231 
231 

*33 
233 
234 

0.190 
.191 
.192 
.193 
.194 

O.OO2  3199 
.002  3460 
.OO2  3722 
.002  3985 
.002  4251 

26l 
262 
263 
266 
267 

o.ooi  8602 
.001  8789 
.001  8976 
.001  9165 
.001  9354 

187 
187 
189 
189 
190 

0.250 
.251 
.252 

•*53 
.254 

0.004  1835 
.004  2199 
.004  2566 
.004  2934 
.004  3305 

364 

36? 
368 
371 
372 

0.003  i*45 
.003  1480 
.003  1716 
.003  1952 
.003  2189 

235 
236 
236 

*37 
238 

0.195 
.196 
.197 
.198 
.199 

0.002  45l8 
.002  4786 
.002  5056 
.002  5328 
.002  5602 

268 
270 
272 
274 
275 

o.ooi  9544 
.001  9735 
.001  9926 
.002  0119 

.002  0312 

191 
191 
'93 
193 
J95 

0.255 
.256 
.257 
.258 
.259 

0.004  3677 
.004  4051 
.004  4427 
.004  4804 
.004  5184 

374 
376 

377 
380 
382 

0.003  2427 
.003  2666 
.003  2905 
.003  3146 
.003  3387 

239 
239 
241 
241 
241 

0.200 
.201 
.202 
.203 
.204 

O»OO2  5877 
.002  6154 
.002  6433 
.002  6713 
.002  6995 

277 
279 
280 
282 
283 

0.002  0507 
.002  0702 
.002  0897 
.OO2  1094 
.OO2  1292 

'95 
195 
197 
198 
198 

0.260 
.261 
.262 
.263 
.264 

0.004  5566 
.004  5949 
.004  6334 
.004  6721 
.004  7111 

383 

385 
387 
390 
391 

0.003  3628 
.003  3871 
.003  4114 
.003  4158 
.003  4603 

243 

243 
244 
245 
245 

0.205 
.206 
.207 
.208 
.209 

0.002  7278 
.002  7564 
.002  7851 
.002  8139 
.002  8429 

286 
287 
288 
290 
293 

O.O02  1490 
.002  1689 
.002  1889 
.OO2  2090 
.002  2291 

199 

200 
201 
2OI 
203 

0.265 
.266 
.267 
.268 
.269 

0.004  7502 
.004  7894 
.004  8289 
.004  8686 
.004  9085 

392 

395 
397 
399 
400 

0.003  4848 
.003  5094 
.003  5341 
.003  5589 
.003  5838 

246 
247 
248 
249 
249 

O.2IO 
.211 
.212 
.213 
.214 

O.002  8722 
.002  9015 
.002  9311 
.002  9608 
.002  9907 

a93 
296 
297 
299 

300 

0.002  2494 
.002  2697 
.OO2  2901 
.002  3106 
.OO2  3311 

203 
204 
205 
205 
207 

0.270 
.271 
.272 
.273 
.274 

0.004  9485 
.004  9888 
.005  0292 
.005  0699 
.005  1107 

4°3 
404 
407 
408 
410 

0.003  6087 
.003  6337 
.003  6587* 
.003  6839 
.003  7091 

250 
250 
252 
252 
*53 

O.2I5 
.216 
.217 
.218 
.219 

0.003  0207 
.003  0509 
.003  0814 
.003  III9 
.003  1427 

302 
305 

3°5 
308 
309 

O.002  3518 
.002  3725 
.002  3932 
.OO2  4142 
.002  4352 

207 
207 
2IO 
210 
210 

0.275 
.276 
.277 
.278 
.279 

0.005  1517 
.005  1930 
.005  2344 
.005  2760 
.005  3178 

413 
414 
416 
418 
420 

0.003  7344 
.003  7598 
.003  7852 
.003  8107 
.003  8363 

254 
254 

*55 
256 

257 

O.22O 
.221 
.222 
.223 
.224 

0.003  1736 
.003  2047 
.003  2359 
.003  2674 
.003  2990 

3" 
312 

3'5 
316 
318 

O.OO2  4562 

.002  4774 
.002  4986 

.002  5199 
.OO2  5412 

212 
212 
213 
2I3 
215 

0.280 
.281 
.282 
.283 
.284 

0.005  3598 
.005  4020 
.005  4444 
.005  4870 
.005  5298 

422 

4*4 
426 
428 
430 

0.003  8620 
.003  8877 
.003  9135 
.003  9394 
.003  9654 

% 

III 

260 

O.225 
.226 
.227 
.228 
.229 

0.003  3308 
.003  3627 

.003  3949 
.003  4272 
.003  4597 

3*9 

322 

3*3 

3*5 
327 

O.002  5627 
.002  5842 
.OO2  6058 
.002  6275 
.002  6493 

"I 

216 

217 
2Ig 

218 

0.285 
.286 
.287 
.288 
.289 

0.005  5728 
.005  6160 
.005  6594 
.005  7030 
.005  7468 

432 

434 
436 
438 
44° 

0.003  9914 
.004  0175 
.004  0437 
.004  0700 
.004  0963 

261 

262 
263 
263 
264 

0.230 
.231 
.232 
.233 
•234 

0.003  4924 
.003  5252 
.003  5582 
.003  5914 
.003  6248 

328 

330 
33* 
334 
336 

0.002  6711 
.OO2  6931 
.002  7151 
.002  7371 

.002  7593 

22O 
220 
220 
222 
223 

0.290 
.291 
.292 
.293 
.294 

0.005  7908 
.005  8350 
.005  8795 
.005  9241 
.005  9689 

44* 
445 
446 

448 
45° 

0.004  1227 
.004  1491 
.004  1757 
.004  2023 
.004  2290 

26^ 
266 
266 

^  \ 
267 

0.235 
.236 

•*37 
.238 
.239 

0.003  65^4 
.003  6921 
.003  7260 
.003  7601 
.003  7944 

337 
339 
341 

343 

•7  AC 

0.002  7816 
.002  8039 
.002  8263 
.002  8487 
.002  8713 

223 
224 
224 
226 
226 

0.295 
.296 
.297 
.298 
.299 

0.006  0139 
.006  0591 
.006  1045 
.006  1502 
.006  1960 

452 
454 
457 
458 
461 

0.004  2557 
.004  2826 
.004  3095 
.004  3364 
.004  3635 

269 
269 
269  \ 
271  , 
271 

0.240 

0.003  8289 

J  i  J 

0.002  8939 

0.300 

0.006  2421 

0.004  3906 

631 


TABLE  XV. 

For  Elliptic  Orbits  of  great  eccentricity. 


c  or  3 

log  .B0  or  log  .B0' 

Diff. 

log^V 

Diff. 

eorS 

logj&oorlog.Bo' 

Diff. 

log  N 

Diff. 

0 

0 

0 

o.ooo  oooo 

o.ooo  oooo 

30 

o.ooo  0066 

o.ooo  6400 

A  •>& 

1 

.000  oooo 

o 

Q 

.000  0007 

7 

2  1 

31 

.000  0075 

9 

J  J 

.000  6836 

436 

2 

.000  OOOO 

.000  0028 

tft 

32 

.000  0086 

.000  7286 

450 

3 

.000  oooo 

o 
o 

.000  0064 

3b 

40 

33 

.000  0097 

I  I 

12 

.000  7750 

464 

4 

.000  OOOO 

o 

.000  0113 

64 

34 

.000  0109 

13 

.000  8229 

493 

5 

o.ooo  oooo 

o 

o.ooo  0177 

78 

35 

0.000  0122 

15 

o.ooo  8722 

508 

6 

.000  OOOO 

o 

.000  0255 

Q2 

36 

.000  0137 

.000  9230 

J 

7 

.000  OOOO 

o 

.000  0347 

y 

107 

37 

.000  0153 

18 

.000  9753 

537 

8 

.000  oooo 

.000  0454 

1  20 

38 

.000  0171 

IQ 

.001  0290 

J  J  / 

5r  2 

9 

.000  OOOI 

o 

.000  0574 

39 

.000  0190 

*•  7 

20 

.001  0842 

* 

567  1 

10 

O.OOO  OOOI 

o 

o.ooo  0709 

149 

40 

O.OOO  O2  1  0 

22 

o.oo  i  1409 

581 

11 
12 

.000  OOOI 
.000  0002 

I 

0 

.000  0858 

.000  I  02  I 

163 

178 

41 
42 

.000  0232 
.000  0255 

11 

.001  1990 
.001  2586 

596 

611 

13 

.OOO  OOO2 

I 

.000  1199 

43 

.000  0281 

27 

.001  3197 

626 

14 

.000  0003 

I 

.000  1390 

206 

44 

.000  0308 

/ 

29 

.001  3823 

640 

15 
16 

o.ooo  0004 
.000  0005 

I 

2 

o.ooo  1596 
.000  1816 

220 
27C 

45 
46 

o.ooo  0337 
.000  0368 

31 

77 

o.oo  i  4463 
.001  5118 

655 

670 

17 

.000  0007 

2 

.000  2051 

JJ 

248 

47 

.000  0401 

36 

.001  5788 

685 

18 

.000  0009 

2 

.000  2299 

Tr 

263 

48 

.000  0437 

38 

.001  6473 

J 

700 

19 

.000  00  1  1 

2 

.000  2562 

277 

49 

.000  0475 

J 

40 

.001  7173 

/  w 

715 

20 
21 
22 
23 
24 

o.ooo  0013 
.000  0016 
.000  0019 
.000  0023 
.000  0027 

3 
3 
4 
4 
5 

o.ooo  2839 
.000  3131 
.000  3437 
.000  3757 
.000  4091 

292 
306 

320 

334 
349 

50 
51 
52 
53 
54 

o.ooo  0515 
.000  0558 
.000  0604 
.000  0652 
.000  0703 

3 

48 

51 

54 

o.ooi  7888 

.001  86l8 

.001  9362 

.002  0122 
.002  0897 

73° 
744 
760 

775 
79° 

25 
26 
27 

28 

o.ooo  0032 
.000  0037 
.000  0043 
.000  0050 

•1 

7 
7 

o.ooo  4440 
.000  4803 
.000  5181 
.000  5573 

367 

378 

392 

4°  7 

55 
56 
57 

58 

o.ooo  0757 
.000  0815 
.000  0875 
.000  0939 

64 

68 

0.002  1687 

.002  2493 

.002  3313 
.002  4149 

806 
820 

836 

851 

29 

.000  0057 

9 

.000  5980 

420 

59 

.000  1007 

.002  5000 

866 

30 

o.ooo  0066 

o.ooo  6400 

60 

o.ooo  1078 

0.002  5866 

TABLE  XVI. 

For  Hyperbolic  Orbits. 


m  or  ?! 

log  Q  or  log  Qf 

log  I.  Diff. 

log  half  II.  Diff. 

m  orra 

log  Q  or  log  Q' 

log  I.  Diff. 

log  half  II.  Diff. 

0.00 
.01 
.02 

.04 

0.000    OOOO 

9-999  987° 
•999  9479 
.999  8828 

•999  79J7 

2.41597* 
2.71675* 
2.89259* 
3.01741* 

2-1149* 
2.1146* 

2.1130* 

O.IO 
.11 
.12 

•'3 
.14 

9.998   7021 
.998   4308 
.998    1342 
•997  8123 
•997  4654 

3.41256* 
3.45326* 
3.49028* 
3.52423* 
3-55547» 

2.1046* 

a.  1  025* 

2.1003* 
2.0978* 
2.0952* 

0.05 

.06 
.07 
.08 
.09 

9  999  6746 
•999  53i6 
.999  3628 
.999  1682 
.998  9479 

3.19290* 

3.31687* 
3-36745« 

2.II2I* 
2.1  1  IO* 

2.1097* 
2.I082* 
2.1065* 

0.15 
.17 
.19 

9.997  0936 
.996  6971 
.996  2760 
•995  8305 
•995  36°8 

3-58453» 
3.61154* 
3.67679* 
3.66048* 
3.68276* 

2.0923* 

2.0892* 
2.0860* 
2.0826* 
2.0790* 

O.IO 

9.998  7021 

3.41256* 

2.1046* 

O.2O 

9.994  8671 

3-7°378n 

2.0"C2n 

632 


TABLE  XVII, 

For  special  Perturbations. 


q,  <f,  «" 

For  positive  values  of  the  Argument. 

For  negative  values  of  the  Argument. 

log/ 

Diff. 

log/',  log/" 

Diff. 

log/ 

Diff. 

log/',  log/" 

Diff. 

o.oooo 

0.477  1213 

1086 

0.301  0300 

860 

0.477  1213 

1086 

0.301  0300 

860 

.0001 

.477  0127 

1085 

.300  9431 

u\jy 

868 

•477  2299 

1086 

.301  1169 

ovsy 

868 

.0002 

.0003 
.0004 

.476  9042 
.476  7957 
.476  6872 

1085 
1085 
1085 

.300  8563 

.300  7695 

.300  6827 

868 
868 
868 

•477  33»5 
•477  447i 
•477  5558 

1086 
1087 
1087 

.301  2037 

.301  2906 

.301  3776 

869 
870 
869 

0.0005 
.0006 

.0007 
.0008 
.0009 

0.476  5787 
.476  4702 
.476  3618 

•476  »534 
.476  1450 

1085 
1084 
1084 
1084 
1083 

0.300  5959 
.300  5092 
.300  4224 

•3°°  3357 
.300  2490 

867 
868 
867 
867 
867 

0.477  6645 

•477  77  32 
.477  8819 
.477  9906 
.478  0994 

1087 
1087 
1087 
1088 
1088 

0.301  4645 
•301  5515 

.301  6384 

.301  7254 

.301  8124 

870 
869 

870 
870  . 
871 

O.OOIO 
.0011 
.0012 

.0013 
.0014 

0.476  0367 
.475  9284 
.475  8201 
•475  7u8 
•475  6°35 

1083 
1083 
1083 
1083 
1082 

0.300  1623 
.300  0756 
.299  9889 
.299  9023 
.299  8157 

867 
867 
866 
866 
866 

0.478  2082 
.478  3170 
•478  4^59 
•478  5348 
•478  6437 

1088 
1089 
1089 
1089 
1089 

0.301  8995 
.301  9865 
.302  0736 
.302  1606 

.302  2477 

870 
871 
870 
871 
871 

0.0015 
.0016 
.0017 
.0018 
.0019 

o-475  4953 
•475  387i 
-475  2789 
.475  1707 
.475  0626 

1082 
1082 
1082 
1081 
1081 

0.299  729* 
.299  6425 

•^99  5559 
.299  4693 
.299  3828 

866 
866 
866 
865 
865 

0.478  7526 
.478  8615 
.478  9705 
•479  °795 
•479  1885 

1089 
1090 
1090 
1090 
1090 

0.302  3348 

.302  4220 
.302  5091 

.302  5963 
.302  6835 

872 
871 
872 
872 
872 

0.0020 
.0021 
.0022 
.0023 
.0024 

0-474  9545 
.474  8464 

•474  7383 
•474  63°3 
•474  5"  3 

1081 

1081 
1080 
1080 
1080 

0.299  2963 
.299  2098 
.299  1237 
.299  0368 
.298  9504 

865 
865 
865 
864 
865 

0.479  a975 
•479  4°65 
.479  5156 
.479  6247 
•479  7338 

1090 
1091 
1091 
1091 
1092 

0.302  7707 
.302  8579 
.302  9451 
.303  0324 

.303  1196 

872 
872 

873 
872 
873 

O.0025 
.0026 
.OO27 
.0028 
.0029 

0.474  4H3 
.474  3063 

•474  1983 
.474  0904 

•473  98*5 

1080 
1080 
1079 
1079 
1079 

0.298  8639 
.298  7775 
.298  6911 
.298  6047 
.298  5184 

864 
864 
864 
863 
864 

0.479  8430 

•479  9522 
.480  0614 
.480  1706 
.480  2798 

1092 
1092 
1092 
1092 
1093 

0.303  2069 

.303  2942 
.303  3815 
.303  4689 

•3°3  5562 

873 
873 
874 
*73 
874 

0.0030 
.0031 
.0032 
•0033 
.0034 

0.473  8746 
•473  7667 
•473  6589 
•473  55" 
•473  4433 

1079 

1078 
1078 
1078 
1078 

0.298  4320 
.298  3457 
.298  2594 
.298  1731 
.298  0868 

863 
863 
863 
863 
863 

0.480  3891 
.480  4984 
.480  6077 
.480  7170 
.480  8264 

1093 
1093 
1093 
1094 
1094 

0.303  6436 
.303  7310 
.303  8184 
.303  9058 
•3°3  9933 

874 
874 

874 
875 
874 

0.0035 
.0036 
.0037 
.0038 
.0039 

o-473  3355 
•473  "78 
•473  "ox 
•473  0^4 
.472  9047 

1077 

1077 
1077 
1077 
1077 

0.298  0005 
.297  9143 
.297  8280 
.297  7418 
.297  6556 

862 
863 
862 
862 

861 

0.480  9358 
.481  0452 
.481  1547 
.481  2641 
.481  3736 

1094 
1095 
1094 
1095 
1095 

0.304  0807 
.304  1682 

•3°4  ^557 
.304  3432 
.304  4308 

875 
875 
875 
876 

875 

0.0040 
.0041 
.0042 
.0043 
.0044 

0.472  7970 
.472  6894 
.472  5818 
.472  4742 
.472  3666 

1076 
1076 
1076 
1076 
1075 

0.297  5695 
.297  4833 
.297  3972 
.297  3110 
.297  2249 

862 
861 
862 
861 
861 

0.481  4831 
.481  5926 
.481  7022 
.481  8118 
.481  9214 

1095 
1096 
1096 
1096 
1096 

0.304  5183 
.304  6059 
.304  6935 
.304  7811 
.304  8687 

876 
876 
876 
876 

876  ' 

0.0045 
.0046 
.0047 
.0048 
.0049 

0.472  2591 
.472  1516 
.472  0441 
.471  9366 
.471  8292 

1075 
1075 
1075 
1074 
1074 

0.297  1388 
.297  0528 
.296  9667 
.296  8807 
.296  7946 

860 
861 
860 
861 
860 

0.482  0310 
.482  1407 
.482  2504 
.482  3601 
.482  4698 

1097 
1097 
1097 
1097 
1098 

0.304  9563 
.305  0440 
.305  1317 
.305  2194 
.305  3071 

877 
877 
877 

!?? 

i  0.0050 
.0051 
.0052 
.0053 
.0054 

0.471  7218 
.471  6144 
.471  5070 
.471  3996 
.471  2923 

1074 
1074 
1074 
1073 
1073 

0.296  7086 
.296  6226 
.296  5367 
.296  4507 
.296  3648 

860 

859 
860 
859 
860 

0.482  5796 
.482  6894 
.482  7992 
.482  9090 
.483  0188 

1098 
1098 
1098 
1098 
1099 

0.305  3948 
.305  4825 

•3°5  5703 
.305  6581 

•3°5  7459 

Ill 

878 
8?8 
878 

0.0055 
.0056 
.0057 
.0058 
.0059 
.0060 

0.471  1850 
.471  0777 
.470  9704 
.470  8632 
.470  7560 
.470  6488 

1073 

1073 
1072 
1072 
1072 

0.296  2788 
.296  1929 
.296  1070 

.296  0212 

•^95  9353 
.295  8495 

859 

III 
III 

0.483  1287 
.483  2386 

•483  3485 
.483  4584 
.483  5684 
.483  6784 

1099 
1099 
1099 

IIOO 
I  100 

0.305  8337 

•3°5  9*'5 
.306  0094 
.306  0973 
.306  1851 
.306  1730 

878 
879 

879 
878 
879 

633 


TABLE  XVII, 

For  special  Perturbations. 


ft  tf,  2" 

For  positive  values  of  the  Argument. 

For  negative  values  of  the  Argument. 

log/ 

Diff. 

log/',  log/" 

Diff. 

log/ 

Diff. 

log/',  log/" 

Diff. 

0.0060 
.0061 
.0062 
.0063 
.0064 

0.470  6488 
.470  5416 
.470  4345 
.470  3274 
.470  2203 

1072 
1071 
1071 
1071 
1071 

0.295  8495 
.295  7637 
.295  6779 
.295  5921 
.295  5063 

858 
858 
858 
858 
858 

0.483  6784 
.483  7884 
.483  8984 
.484  0085 
.484  1186 

1  100 
1  100 
HOI 
HOI 
HOI 

0.306  2730 
.306  3610 

.306  4489 
.306  5369 

.306  6248 

880 
879 
880 

III 

0,0065 
.0066 
i  .0067 
.0068 
.0069 

0.470  1132 
.470  0062 
.469  8992 
.469  7922 
.469  6852 

1070 
1070 
1070 
1070 
1070 

0.295  4205 
.295  3348 
.295  3491 
.295  1634 
.295  0777 

857 
857 

857 
857 
857 

0.484  2287 
.484  3388 
.484  4490 
.484  5592 
.484  6694 

HOI 
1102 
1102 
1  102 
1102 

0.306  7128 
.306  8009 
.306  8889 
.306  9769 

.307  0650 

881 
880 
880 
881 
881 

0.0070 
.0071 
.0072 
.0073 
.0074 

0.469  5782 
.469  4713 
.469  3644 
.469  2575 
.469  1506 

1069 
1069 
1069 
1069 
1069 

0.294  9920 
.294  9064 
.294  8208 
.294  7351 
.294  6495 

856 
856 

857 
856 

855 

0.484  7796 
.484  8898 
.485  oooi 

.485  1104 

.485  2207 

II  O2 
1103 
1103 
1103 
HO4 

0.307  1531 
.307  2412 
.307  3293 
.307  4174 

.307  5056 

881 
881 
881 
882 
882 

0.0075 
.0076 
.0077 
.0078 
.0079 

0.469  0437 
.468  9369 
.468  8301 
.468  7233 
.468  6165 

1068 
1068 
1068 
1068 
1067 

0.294  5640 
.294  4784 
.294  3928 
.294  3077 
.294  2218 

856 
856 
855 
855 
855 

0.485  3311 
.485  4415 
.485  5519 

.485  6623 

.485  7728 

1104 
1104 
1104 
1105 
1105 

0.307  5938 

.307  6820 

.307  7702 
.307  8584 
.307  9466 

882 

882 
882 
882  ! 
883 

0.0080 
.0081 
.0082 
.0083 
.0084 

0.468  5098 
.468  4031 
.468  2964 
.468  1897 
.468  0831 

1067 
1067 
1067 
1066 
1066 

0.294  1363 
.294  0508 
.293  9653 
.293  8799 
•*93  7945 

855 
855 
854 

854 
855 

0.485  8833 
.485  9938 

.486  1043 
.486  2149 

.486  3255 

1105 
1105 

1106 
1106 
1106 

0.308  0349 

.308  1232 
.308  2115 
.308  2998 
.308  3881 

883 

883  | 
883 
883 
884 

0.0085 
.0086 
.0087 
.0088 
.0089 

0.467  9765 
.467  8699 
.467  7633 
.467  6567 
,467  5502 

1066 
1066 
1066 
1065 
1065 

0.293  709° 
.293  6236 

•293  5383 
.293  4529 
.293  3675 

854 
853 
854 

854 
853 

0.486  4361 

.486  5467 
.486  6573 

.486  7680 
.486  8787 

1106 
1106 

1107 
1107 
1107 

0.308  4765 
.308  5648 

.308  6532 
.308  7416 
.308  8301 

883 
884 
884 
885 
884 

0.0090 
.0091 
.0092 
.0093 
.0094 

0.467  4437 
.467  3372 
.467  2307 
.467  1243 
.467  0179 

1065 
1065 
1064 
1064 
1064 

0.293  2822 
.293  1969 
.293  1116 
.293  0263 
.292  9411 

853 
853 
853 
852 

853 

0.486  9894 
.487  iooi 
.487  2109 

.487  3217 
.487  4325 

1107 
1108 
1108 
1108 
1108 

0.308  9185 
.309  0070 

.309  0954 
.309  1839 
.309  2725 

885 
884 
885 
886 
885 

0.0095 
.0096 
.0097 
.0098 
.0099 

0.466  9115 
.466  8051 
.466  6988 
.466  5925 
.466  4862 

1064 
1063 
1063 
1063 
1063 

0.292  8558 
.292  7706 
.292  6854 
.292  6002 
.292  5150 

852 

85* 
852 
852 

0.487  5433 
.487  6542 
.487  7651 
.487  8760 
.487  9869 

1109 
1109 
1109 
1109 

i  no 

0.309  3610 
.309  4495 
.309  5381 
.309  6267 
.309  7153 

885 
886 
886 
886 
886 

O.OIOO 
.0101 
.0102 
.0103 
.0104 

0.466  3799 
.466  2736 
.466  1674 
.466  0612 
.465  9550 

1063 
1062 
1062 
1062 
1062 

0.292  4298 
.292  3447 
.292  2595 
.292  1744 
.292  0893 

851 

852 
851 

851 
850 

0.488  0979 
.488  2089 
.488  3199 
.488  4309 
.488  5420 

mo 

IHO 
IHO 

II  1  1 
II  II 

0.309  8039 
.309  8926 
.309  9812 
.310  0699 
.310  1586 

887 
886 
887 
887 
887 

O.OI05 
.OI06 
.0107 
.0108 
!   .0109 

0.465  8488 
.465  7427 
.465  6366 

•465  53°5 
.465  4244 

106^, 
1061 
1061 
1061 
1061 

0.292  0043 
.291  9192 
.291  8341 
.291  7491 
.291  6641 

851 

851 
850 
850 
850 

0.488  6531 
.488  7642 
.488  8753 
.488  9865 
.489  0977 

I  II  I 
I  III 

I  I  12 
HI2 
III2 

0.310  2473 
.310  3360 
.310  4248 
.310  5136 
.310  6023 

887 
888 
888 
887 
888 

O.OIIO 
.0111 
.OI  12 

0.465  3183 
.465  2123 
.465  1063 

1060 
1060 
1060 

0.291  5791 
.291  4941 
.291  4092 

850 

849 
850 

0.489  2089 
.489  3201 
.489  4314 

III2 
III3 

0.310  6911 
.310  7800 
.310  8688 

889 
888 
889 

.0113 
•OH4 

.465  0003 
.464  8943 

1060 
1059 

.291  3242 
.291  2393 

849 
849 

.489  5427 
.489  6540 

III3 

.310  9577 
.311  0465 

888 
889 

O.OII5 
.OIl6 
.OII7 
,OIl8 
0119 
.OI20 

0.464  7884 
.464  6825 
.464  5766 
.464  4707 
.464  3648 
.464  2590 

1059 
1059 
1059 
1059 
1058 

0.291  1544 
.291  0695 
.290  9846 
.290  8997 
.290  8149 
.290  7300 

849 

849 
849 

0.489  7653 
.489  8767 
.489  9881 
.490  0995 
.490  2109 
.490  3223 

IH4 

III4 
IH4 
III4 

0.311  1354 
.311  2243 
.311  3133 
.311  4022 
.311  4912 
.311  5802 

889 
890 

889 
890 
890 

634 


TABLE  XVIL 

For  special  Perturbations. 


q,  f,  a" 

For  positive  values  of  the  Argument. 

For  negative  values  of  the  Argument. 

log/ 

Diff. 

log/',  log/" 

Diff. 

log/ 

Diff. 

log/',  log/" 

Diff. 

0.0120 
.0121 
.OI22 
.0123 
.0124 

0.464  2590 
.464  1532 
.464  047^. 
.463  9416 
.463  8359 

1058 
1058 
1058 
1057 
1057 

0.290  7300 

.290  6452 

.290  5604 

.290  4756 
.290  3909 

848 
848 
848 

III 

0.490  3223 
.490  4338 

•49°  5453 
.490  6568 
.490  7684 

III5 

III5 

"^ 

uto 

ii  16 

0.3II  5802 
.311  6692 
.311  7582 
.311  8472 
.3II  9363 

890 
890 
890 
89, 
891 

O.OI25 
.0126 
.0127 

;  .0128 
.0129 

0.463  7302 
.463  6245 
.463  5188 
.463  4132 
.463  3076 

1057 
1057 
1056 
1056 
1056 

0.290  3061 
.290  2214 
.290  1367 
.290  0520 

.289  9673 

847 
847 
847 
847 
847 

0.490  8800 
.490  9916 
.491  1032 
.491  2149 
.491  3266 

1  1  16 
1116 
1117 
1117 
1117 

0.312  0254 
.312  1145 
.312  2036 
.312  2927 
.312  3819 

89I 
891 
891 
892 
891 

:   O.OI3O 
.0131 
.OI32 
.0133 
.0134 

0.463  2020 
.463  0964 
.462  9908 
.462  8853 
.462  7798 

1056 
1056 
1055 
1055 
1055 

0.289  8826 
.289  7980 

.289  7134 

.289  6287 
.289  5441 

846 
846 
847 
846 
845 

0.491  4383 
.491  5500 
.491  6618 
.491  7736 
.491  8854 

1117 
1118 
1118 
1118 
1118 

0.312  4710 
.312  5602 
.312  6494 
.312  7387 
.312  8279 

892 
892 

893 
892 

893 

0.0135 
.0136 

•0137 
.0138 
.0139 

0.462  6743 
.462  5688 
.462  4633 

.462  3579 
.462  2525 

1055 

i°55 
1054 
1054 
1054 

0.289  4596 
.289  3750 
.289  2904 
.289  2059 
.289  1214 

846 
846 
845 
845 
845 

0.491  9972 
.492  1091 

.492  2210 
.492  3329 
.492  4448 

1119 
1119 
1119 
1119 
1119 

0.313  9172 
.313  0064 

•313  °957 
.313  I850 
.313  2744 

892 

893 
893 

894 
893 

0.0140 
.0141 
.0142 
.0143 
.0144 

0.462  1471 
.462  0417 
.461  9364 
.461  8311 
.461  7258 

1054 
1053 
1053 
1053 
i°53 

0.289  0369 
.288  9524 
.288  8679 
.288  7835 
.288  6990 

845 
845 
844 
845 
844 

0.492  5567 
.492  6687 
.492  7807 
.492  8927 

•493  o°47 

I  120 
1120 
1  1  20 
1120 

I  121 

0.313  3637 

•3'3  453i 
•3'3  54*5 
.313  6319 
.313  7213 

894 
894 
894 

894 
895 

0.0145 
.0146 
.OI47 
.0148 
.0149 

0.461  6205 
.461  5153 
.461  4101 
.461  3049 
.461  1997 

1052 
1052 
1052 
1052 
1052 

0.288  6146 
.288  5302 
.288  4458 
.288  3615 
.288  2771 

844 

844 

843 
844 

843 

0.493  I1[68 
•493  "89 
•493  3410 
•493  45  3* 
•493  5654 

II2I 
II2I 
1  1  22 
1122 
1  1  22 

0.313  8108 
.313  9002 
.313  9897 
.314  0792 
.314  1687 

894 
895 
895 

III 

0.0150 
.0151 
.0152 
.0153 
.0154 

0.461  0945 
.460  9894 
.460  8843 
.460  7792 
.460  6741 

1051 
1051 
1051 
1051 
1051 

0.288  1928 
.288  1085 
.288  0241 
.287  9399 
.287  8556 

843 
844 
842 
843 
843 

0.493  6776 

•493  7898 
.493  9021 

•494  oi44 
.494  1267 

1  1  22 
1123 
1123 
1123 
II23 

0.314  2583 
.314  3478 

•3*4  4374 
.314  5270 
.314  6166 

III 
896 
896 
896 

0.0155 
.0156 
.0157 
.0158 
.0159 

0.460  5690 
.460  4640 
.460  3590 
.460  2540 
.460  1490 

1050 
1050 
1050 
1050 
1049 

0.287  7713 
.287  6871 
.287  6029 
.287  5187 
.287  4345 

842 

842 
842 
842 
842 

0.494  2390 

•494  35H 
.494  4638 

•494  5762 
.494  6886 

II24 
1124 
1124 
1124 
II24 

0.314  7062 

•3*4  7959 
.314  8855 

•3*4  975* 
.315  0649 

897 
896 

897 
897 

897 

0.0160 
.0161 
.0162 
.0163 
.0164 

0.460  0441 
•459  939* 
•459  8343 
•459  7294 
•459  6*45 

1049 
1049 
1049 
1049 
1048 

0.287  35°3 
.287  2661 
.287  1820 
.287  0979 
.287  0138 

842 
841 
841 
841 
841 

0.494  8010 

•494  9'35 
.495  0260 

•495  1385 
•495  ^510 

1125 
1125 
1125 
1125 
1126 

0.315  1546 
.315  2444 

•3*5  334i 
.315  4239 

•3*5  5'37 

898 

897 
898 
898 
898 

0.0165 
.0166 
.0167 
.0168 
.0169 

o-459  5*97 
.459  4149 

•459  3101 
•459  2°53 
.459  1006 

1048 
1048 
1048 
1047 
1047 

0.286  9297 
.286  8456 
.286  7615 
.286  6775 
.286  5935 

841 

841 
840 
840 
840 

0.495  3636 
.495  4762 
.495  5888 

•495  7OI5 
.495  8142 

1126 
1126 
1127 
1127 
1127 

0.315  6035 

•3*5  6934 
.315  7832 

•3*5  8731 
.315  9630 

|»| 

898 
899 
899 

899  ! 

0.0170 
.0171 
.0172 
.0173 
.0174 

0.458  9959 
.458  8912 
.458  7865 
.458  6818 
•458  577* 

1047 
1047 
1047 
1046 
1046 

0.286  5095 
.286  4255 
.286  3415 
.286  2575 
.286  1736 

840 
840 
840 

839 
840 

0.495  9*69 
.496  0396 
,496  1524 
.496  2652 
.496  3780 

1127 
1128 
1128 
1128 
1128 

0.316  0529 
316  1428 
.316  2327 
.316  3227 
.316  4127 

899 
899 
900 
900 
900 

0.0175 
.0176 
.0177 
.0178 
.0179 
.0180 

0.458  4726 
.458  3680 
.458  2614 
.458  i5§9 
.458  0544 
•457  9499 

1046 
1046 
1045 
1045 
1045 

0.286  0896 
.286  0057 
.285  9218 
.285  8380 
.285  7541 
.285  6702 

839 
839 
838 

839 
839 

0.496  4908 
.496  6037 
.496  7166 
.496  8295 
.496  9424 
•497  °554 

1129 
1129 
1129 
1129 
1130 

0.316  5027 
.316  5927 
.316  6827 
.316  7728 
.316  8629 
.316  9530 

900 
900 
901 
901 
901 

635 


TABLE  XVII. 

For  special  Perturbations. 


ft  tf>  «" 

For  positive  values  of  the  Argument. 

For  negative  values  of  the  Argument. 

log/ 

Diff. 

log/',  log/" 

Diff. 

log/ 

Diff. 

log/',  log/" 

Diff. 

0.0180 
.0181 
.0182 
.0183 
.0184 

0.457  9499 
•457  8454 
•457  74°9 
•457  6365 
•457  53*1 

1045 
1045 
1044 
1044 

0.285  6702 

.285  5864 

.285  5026 
.285  4188 
•*85  335° 

838 
838 
838 

Q~Q 

0.497  0554 
.497  1684 
.497  2814 
•497  3944 
•497  5°75 

1130 
1130 
1130 
1131 

0.316  9530 
.317  0431 
.317  1332 
.317  2234 
•3*7  3*35 

901 
901 

902 
901 

1044 

o  ^o 

902 

0.0185 
.0186 
.0187 
.0188 
.0189 

0-457  4*77 
•457  3*33 
•457  *i8g 
.457  1146 

•457  OI03 

1044 
1044 
1043 
1043 
1043 

0.285  2512 
.285  1675 
.285  0838 
.285  oooo 
.284  9163 

837 

837 
838 
837 
837 

0.497  6206 
•497  7337 
.497  8468 
•497  9600 
.498  0732 

1131 
1131 
1132 
1132 
"3* 

0.317  4037 

•3*7  4939 
.317  5841 
.317  6744 
.317  7646 

902 
902 

9°3 
902 
903 

0.0190 
.0191 
.0192 
.0193 
.0194 

0.456  9060 
.456  8017 
.456  6975 

•456  5933 
.456  4891 

1043 
1042 
1042 
1042 
1042 

0.284  8326 
.284  7490 

.284  6653 

.284  5817 
.284  4981 

836 

837 
836 
836 

836 

0.498  1864 
.498  2996 
.498  4129 
.498  5262 
.498  6395 

1132 
"33 
"33 
"33 
"33 

0.317  8549 

•3*7  945* 
.318  0355 
.318  1259 
.318  2162 

9°3 

9°3 
904 

9°3 
904 

0.0195 
.0196 
.0197 
.0198 
.0199 

0.456  3849 
.456  2808 
.456  1767 
.456  0726 
•455  9685 

1041 
1041 
1041 
1041 
1041 

0.284  4'45 
.284  3309 
.284  2473 
.284  1637 
.284  0802 

836 
836 
836 

835 
835 

0.498  7528 
.498  8662 
.498  9796 
•499  0930 
•499  *°64 

"34 
"34 
"34 
"34 
"35 

0.318  3066 
.318  3970 
.318  4874 
.318  5778 
.318  5683 

904 

904 
904 
905 
905 

O.O2OO 
.0201 
.O2O2 
.0203 
.0204 

0.455  8644 
.455  7604 
•455  6564 
•455  55*4 
.455  4484 

1040 
1040 
1040 
1040 
1040 

0.283  9967 
.283  9132 
.283  8297 
.283  7462 
.283  6627 

835 
835 
835 
835 
834 

0.499  3199 

•499  4334 
.499  5469 
.499  6604 
•499  774° 

"35 
"35 
1135 
1136 
1136 

0.318  7588 
.318  8492 
.318  9398 
.319  0303 
.319  1208 

904 
900 
905 
905 
906 

0.0205 
.O2O6 
.0207 
.0208 
.0209 

0.455  3444 
•455  *4Q5 
•455  '366 
•455  °3*7 
.454  9288 

1039 
1039 
1039 
1039 
1039 

0.283  5791 
.283  4958 
.283  4124 
.283  3290 
.283  2456 

834 
834 

834 
833 

0.499  8876 

.500  OOI2 
.500  1149 
.500  2286 
.500  3423 

1136 
"37 
"37 
"37 
"37 

0.319  2114 
.319  3020 
.319  3926 
.319  4832 
•3'9  5738 

906 
906 
906 
906 
9°7 

0.0210 
.0211 
.O2I2 
.0213 
.0214 

0.454  8*49 
.454  7211 
•454  6173 
•454  5*35 
•454  4097 

1038 
1038 
1038 
1038 
1037 

0.283  1623 
.283  0789 
.282  9956 
.282  9123 
.282  8290 

834 
833 
833 

833 
833 

0.500  4560 
.500  5697 
.500  6835 

.500  7973 
.500  9111 

"37 
1138 
1138 
1138 
"39 

0.319  6645 

-V9  755* 
.319  8459 
.319  9366 
.320  0273 

907 

907 
907 
907 
908 

O.02I5 
.02l6 
.0217 
.0218 
1   .0219 

0.454  3060 

•454  *°*3 
.454  0986 

•453  9949 
•453  8912 

1037 
1037 
1037 
1037 
1036 

0.282  7457 
.282  6624 
.282  5792 
.282  4959 
.282  4127 

833 
832 

833 
832 
832 

0.501  0250 
.501  1389 
.501  2528 
.501  3667 
.501  4807 

"39 
"39 
"39 

1140 
1140 

t>.32o  1181 
.320  2088 
.320  2996 
.320  3904 
.320  4813 

907 
908 
908 
909 
908 

0.0220 
.0221 
.0222 
.0223 
.O224 

0.453  7876 
•453  684° 
•453  58°4 
•453  47°8 
•453  3733 

1036 
1036 
1036 

I035 

0.282  3295 
.282  2463 
.282  1631 
.282  0800 
.281  9968 

832 
832 
831 
832 
831 

0.501  5947 
.501  7087 
.501  8227 
.501  9368 
.502  0509 

1140 
1140 
1141 
1141 
1141 

0.320  5721 
.320  6630 

•3*°  7539 
320  8448 

•3*o  9357 

909 
909 
909 
909 
909 

0.0225 
.0226 
.O227 
.0228 
.0229 

0.453  *698 

•453  l663 
.453  0628 

•45*  9593 
•45*  8558 

1035 
1035 

I035 
1035 
1034 

0.281  9137 
.281  8306 
.281  7475 
.281  6644 
.281  5814 

831 

83! 
830 
831 

0.502  1650 
.502  2791 
.502  3933 
.502  5075 
.502  6217 

1141 

1142 
1142 
1142 
"43 

0.321  0266 
.321  1176 
.321  2086 
.321  2996 
.321  3906 

910 

910 
910 
910 
910 

O.0230 
.0231 

.0232 

•o*33 
.0234 

0.452  7524 
.452  6490 

•45*  5456 
.452  4422 

•45*  3389 

"034 
I034 
i°34 
1033 
1033 

0.281  4983 
.281  4153 
.281  3323 
.181  2493 
.281  1663 

830 
830 
830 
830 
830 

0.502  7360 
.502  8503 
.502  9640 

•5°3  0789 
.503  1932 

"43 

"43 
"43 
"43 
"44 

0.321  4816 
.321  5727 
.321  6637 
.321  7548 
.321  8460 

9" 
910 

9" 
912 

9" 

0.0235 
.0236 

•o*37 
.0238 
.0239 
.0240 

0.452  2356 

•45*  13*3 
.452  0290 
.451  9258 
.451  8226 
.451  7194 

1033 
1032 
1032 
1032 

0.281  0833 
.281  0004 
.280  9174 
.280  8345 
.280  7516 
.280  6687 

829 
830 
829 
829 
829 

0.503  3076 
.503  4220 

•5°3  5364 
.503  6508 

•503  7653 
•503  8798 

"44 
"44 
"44 
"45 
"45 

0.321  9371 
.322  0282 
.322  1194 
.322  2106 
.322  3018 
.322  3930 

9" 
912 
912 
912 
912 

636 


TABLE  XVII. 

For  special  Perturbations. 


q,  </,  3" 

For  positive  values  of  the  Argument. 

For  negative  values  of  the  Argument. 

log/ 

Diff. 

log/',  log/" 

Diff. 

log/ 

Diflf. 

log/',  log/" 

Diff. 

0.0240 
.0241 
.0242 
.0243 
.0244 

0.451  7194 
.451  6162 
.451  5130 

•45  1  4°99 
.451  3068 

O  M  M  M  M 
CO  CO  CO  CO  CO 

O  O  O  O  O 

0.280  6687 
.280  5858 

.280  5030 
.280  4201 
.280  3373 

829 
828 
829 
828 
828 

0.503  8798 

•5°3  9943 
.504  1089 
-5°4  2235 
.504  3381 

invovo  ^0  vo 

0.322  3930 
.322  4843 
.322  5756 
.322  6668 
.322  7581 

913 
913 
9I2 
913 
914 

0.0245 
.0246 
.0247 
.0248 
.0249 

0.451  2037 
.451  1006 

•45°  9975 
.450  8945 
.450  7915 

1031 
1031 

1030 
1030 
1030 

0.280  2545 
.280  1717 
.280  0889 
.280  0062 
.279  9234 

828 

828 
827 
828 
827 

0.504  4527 
.504  5674 
.504  6821 
.504  7968 
.504  9115 

"47 
"47 
"47 
"47 
1148 

0.322  8495 
.322  9408 
.323  0322 
.323  1236 
.323  2150 

913 
914 
914 
914 
914 

0.0250 
.0251 
.0252 
.0253 
.0254 

0.450  6885 

•450  5855 
.450  4825 
.450  3796 
.450  2767 

1030 
1030 
1029 
1029 
1029 

0.279  8407 
.279  7580 
.279  6753 
.279  5926 
.279  5099 

827 
827 
827 
827 
826 

0.505  0263 
.505  1411 

•5°5  2559 
.505  3707 
.505  4856 

00  00  00  ON  ON 

0.323  3064 
.323  3978 

•323  4893 
.323  5808 
.323  6723 

914 

9'5 
9*5 
9*5 

0.0255 

0.450  1738 

I  O2  Q 

0.279  4*73 

827 

0.505  6005 

T  TAG 

0.323  7638 

oi  e 

.0256 
.0257 
.0258 
.0259 

.450  0709 
.449  9681 
•449  8653 
•449  7625 

1028 
1028 
1028 
1028 

.279  3446 
.279  2620 
.279  1794 
.279  0968 

O  *»  / 

826 
826 
826 
825 

.505  7154 
.505  8303 

•5°5  9453 
.506  0603 

i  •I'j-y 

"49 
1150 
1150 
1150 

•323  8553 
.323  9469 
.324  0384 
.324  1300 

916 

9'5 
910 
917 

0.0260 
.0261 
.0262 
.0263 
.0264 

0.449  6597 
•449  5569 
•449  4542 
•449  3515 
.449  2488 

1028 
1027 
1027 
1027 
1027 

0.279  OI43 
.278  9317 
.278  8492 
.278  7666 
.278  6841 

826 
825 
826 
825 
825 

0.506  1753 
.506  2903 
.506  4054 
.506  5205 
.506  6356 

1150 
1151 
1151 
1151 
1152 

0.324  2217 

•324  3*33 
.324  4049 
.324  4966 
.324  5883 

916 
916 
917 
917 
917 

0.0265 
.0266 
.0267 
.0268 
.0269 

0.449  J46i 

•449  0435 
.448  9409 
.448  8383 
•448  7357 

1026 
1026 
1026 
1026 
1026 

0.278  6016 
.278  5191 
.278  4367 
.278  3542 
.278  2718 

825 

824 
825 
824 
824 

0.506  7508 
.506  8660 
.506  9813 
.507  0965 
.507  2117 

1152 

"53 
1152 
1152 
"53 

0.324  6800 
.324  7717 
.324  8635 

•3*4  9553 
.325  0470 

917 
918 
918 
917 
919 

0.0270 
.0271 
.0272 
.0273 
.0274 

0.448  6331 

•448  5305 
.448  4280 

•448  3*55 
.448  2230 

1026 
1025 
1025 
1025 
1025 

0.278  1894 
.278  1070 
.278  0246 
.277  9422 
.477  8599 

824 

824 
824 
823 
824 

0.507  3270 
.507  4423 

•5°7  5577 
.507  6731 
.507  7885 

"53 

"54 
"54 
"54 
"54 

0.325  1389 
.325  2307 
.325  3225 
.325  4144 
.325  5063 

918 
918 
919 
919 
919 

0.0275 
.0276 
.0277 
.0278 
.0279 

0.448  1205 
.448  01  8  i 
•447  9*57 
•447  8133 
.447  7109 

1024 
1024 
1024 
1024 
1024 

0.277  7775 
.277  6952 
.277  6129 
.277  5306 
.277  4483 

823 
823 
823 
823 
822 

0.507  9039 
.508  0194 
.508  1349 
.508  2504 
.508  3659 

"55 
"55 
"55 
"55 
"55 

0.325  5982 
.325  6901 
.325  7821 
.325  8740 
.325  9660 

919 
920 
919 
920 
920 

0.0280 
.0281 
.0282 
.0283 
.0284 

0.447  6085 
.447  5062 

•447  4°  3  9 
.447  3016 

•447  2993 

1023 
1023 
1023 
1023 
1023 

0.277  3661 
.277  2838 
.277  2016 
.277  1194 
.277  0372 

823 
822 
822 
822 
822 

0.508  4814 
.508  5970 
.508  7126 
.508  8282 
.508  9439 

1156 
1156 
1156 
"57 
.  "57 

0.326  0580 
.326  1500 
.326  2421 
.326  3341 
.326  4262 

920 
921 
920 
921 

92! 

0.0285 
.0286 
.0287 
.0288 
.0289 

0.447  0970 
.446  9948 
.446  8926 
.446  7904 
.446  6882 

1022 
1022 
1022 
1022 
1021 

0.276  9550 
.276  8728 
.276  7907 
.276  7086 
.276  6264 

822 
821 
821 
822 

821 

0.509  0596 
.509  1753 
.509  2910 
.509  4068 
.509  5226 

"57 
"57 
1158 
1158 
1158 

0.326  5183 
.326  6104 
.326  7026 
.326  7947 
.326  8869 

92I 
922 
92I 
922 
922 

0.0290 
.0291 
.0292 
.0293 
.0294 

0.446  5861 
.446  4840 
.446  3819 
.446  2798 
.446  1777 

1021 
1021 
I  O2  1 
1021 
1021 

0.276  5443 
.276  4622 
.276  3802 
.276  2981 
.276  2161 

821 
820 
821 
820 
821 

0.509  6384 

•5°9  7543 
.509  8702 
.509  9861 

.510  1020 

"59 
"59 
"59 
"59 
"59 

0.326  9791 

.327  0713 
.327  1635 
.327  2558 
.327  3481 

922 
922 
923 
923 
923 

0.0295 
.0296 
.0297 
.0298 
.0299 
.0300 

0.446  0756 

•445  9736 
.445  8716 

•445  7696 
.445  6676 

•445  5657 

1020 
1020 
IO2O 
1020 
IOI9 

0.276  1340 
.276  0520 
.275  9700 
.275  8880 
.275  8061 
.275  7241 

820 
820 
820 
819 
820 

0.510  2179 

•510  3339 
.510  4^.99 
.510  5659 
.510  6819 
.510  7980 

1160 
1160 
1160 
1160 
1161 

0.327  4404 
.327  5327 
.327  6250 
.327  7174 
.327  8097 
.327  9021 

923 
923 
924 
923 
924 

637 


TABLE  XVIII, 

Elements  of  the  Orbits  of  Comets  which  have  been  observed. 


iitfi  lllii 

«£,3sg  -   -    • 


1 

'  fcc , 

:  2' 


1 


3 


r^vO  vo    d 

OCOQOOO  M  vo  vo    Tj-oo 

»$•  m  r~~  co  vn  rj-oo    O  vo    ^t1 

vo  oo   vn  tnoo  ON  ONOO>   t^.  t-^ 

O\  O\  O\  ON  ON  ONONONONON 


oo 


>*   ON  IN 
»0  O     t^ 


d        co  to  O  •*  O 

vo  covo  co  f>  oo  co  IN  •<!-  o  vo  t-x  m  O  t^  oo  O 

Tj-    t~x    t^OO   VO  CO   d    OO     M    VO  OOOt^-INlN  t-^OO 

vn  vnvo  vot^  ONOMOOVO  oo    covo    co  vn  vo    r-- 

t^-OO     CNOO    CN  ^vnONONt-^  OOVnt^-ONt^  OOOO 

ONO\O\O\O\  O\O\O\O\O\  O\O\O\O\ON  O»O\O\CNCN 


oo    •*  vo    co  o    O    rt 

ro  IN  co  c<    rt-  rt  oo 

co  t-  ^J-  co  M    ON  ON 

vo  oo  M   o  vo   rj-  r^ 


ON  ON  ON  ON  ON 


CO  VO 

f~.  00 

q  q  q  q  q    q  q  q  q  q    q  q  q  q  q    q  q  q  q  q    q  q  q  q  q    q  q  q  "ON  q    q  q  q  q  ON 


OOOOO   OOOOO   OOOOO   OOOOO   OOOOO   OOOOO   OOOdvn 


CO  ^ 


vioo    O  vo 


OOQOO^O   OOOOO   OOOOO   OOOOO   OOOOO   OOOOO   OOO-*-vn 


OooMOr^     MONOtn 

^  M          CO  ^-    CO   IN 


vn  O    O    O  oo      ONVO  vo 


oo      ^J-oo    Ovoo      ^vnvnt^co     t~-»r-^coc»t-~.oo    cooo    iNt^.oooorJvnON     o\vndtnON 
vo     ONctONOtnoortcJOiN       toOONWTj-vOco  T^VO    O      oovOcorJ-M       ONl^coclw 


OOOOO   OOOOO   OOOOO   OOOOO   OOOOO   OOOOO   OOOOO 

8Ooo     ooOOO»N      r^HOcoco 
co  rj-     co  rt    H          co     rj- 


to  IN    IN    cooo      vo    co  t—  O\( 
ortvnt^woo      «*4-  vnoo  v 
co  H    cJ    co  co  M    co  cl 


MCOCJVOON      MMMVOOO       OOCOO>NVn      t^rt-ON  O\OO 
^-  VOON     OooOvnTf-      vnMvnOco     IN    r^  H    d    O 

d  d       M    d    co  co 


t^d^  OtnONdON 

O    M    vn  t~x  rj-  ON  ONOO 

vnQco  Mt^-ddO 

MdcoM  ddcoMM 


•gQOOOO  ON-* 
SOOOOO  ooco 
<!«odOvno  vovo 


w    O  oo     vo    O 


CO  i— I  O  O  »O     CO^Ol>TH     O? 
TH  C<J  »O  iO     iO  vO 


ITH^     (N(N».1O1(N    ®1®?(N®1CO     eCWffCCOCO 


638 


TABLE  XVIIL 

Elements  of  the  Orbits  of  Comets  which  have  been  observed. 


1          *•§       t          * 

1*1:11  H$H 

J 


I  1  I 

bJO+J   &C..J   tJO 

Oj  .i     QJ  ,Z^     CD 

MQWOM 


M    SO  Tj-  M    OO    OO 

so    moo    Ort  00    O  so    O    «  OsOs       sort       Os  vo  Os  •*  ON 

u->  u->  m  •<$•  o      vo  O    vo  <*•  m  •<*-  O    O    t^  t-> 

wrowosf-s      COMVOMM  Os  m  Os  ^  O 


so    Os  rj-  Os  m     ON  O 


ON  O    Osoo    Os 


OO  so       O    t^  Os 
x  Osso       O    O 


ro  O 

M    to    00    O         so    t^  00    Os  Osso   "6  t-.  voso    rt  00 

sovo     rx  i-~  rt    m  ON  •*>•  r-~.  ON  O    m 

ON  M       rt    O    O    ^  O  Osso    f^oo  oo 

»o  ONSO    TJ-OO       M    r-x  c<    m  r~*     t-^  M    m  m  M  OsO^hrort 

O>    l~>  t~~  t^.  ON      vo  rt    C^vOO  OO       t^.OO  SO    Oj\  O  O^^    ^°  O^OO 

ot^osONON    osoo'ososON    o^ONO^ONO*  O\O*O\ONO\ 


00 


O    vo  O    t^ 
f^oo    rt    «f 

Os  Os  Os  rt 

rt    Osso  oo 


q  q  q  q  q     a\  q  q  q  q    q  q  q  q  q   vo  o>  cr\  ON  q    q  q  q  q  q    q  q  o  o  o         o  q  o  o 


O       O 

^-^- 


0000 


OO  OO    «J-»  VO 


_     OsOOst~xM        t-^i-it-^ONrO      MSOt^fOOs      HOt^covo      MONrtMON      M 

"so         rtoovoMrtmt^rortt^rtoot^         SOMOOSO      covortMso      rf 


a 


Os  f-»  vo  rf  o     oo    m  vooo    M      M  oo    m  ^so      to  rt   M   tooo      O    O  so   t^  M 

O     rt    cosO  SO    CO      ^  ON  fxOO  OO      OOrtONOstOsOf^vo  t^sO        vo  Os  M  \O    rt 

rt         MMCO  rt  rt    1-1    rt    rt      «    rt         M    rt      to        rt    rt    f 


OO«Ooo      voOOO1'* 


>O  oo    r~»vO    O      M  oo   tooo   vo     On 
Ovo        M    t^  t^     O    M         HM      mt^ 
H         rtMrtmco  MM 


rtrtMsooo 


sOOrt      tooo   rtONM      rt 
sot>.M      comr-t^rt      ^h 


vort      votovoco 


—       ^ 


*s|^I  f|J5§  tsjili  f 

<G$^£  M^^cO^  OS^Q^  £< 


(M  iO  O  CO  CO     t~-  00  00  <M 

OOOOOSOiOS      Oi-Hi-HlO 
LO  LO  iO  vO  iO     CO  «O  SO  «O 


i  00  (M  t-  OO  O  <M  CO 

i  co  t^  t-  t^  oo  oo  ~~ 

i  co  co  co  co  co  co 


639 


TABLE  XVIIL 

Elements  of  the  Orbits  of  Comets  which  have  been  observed. 


§3  -M  +5  *J  ^S 

B>        |  'B    .§         -•    T2  .  d'B 

S      .       3  3      P  c  S  c3     g 

llflH-IFlllllilllll 

.-3045        3  o>  .J3          o  <u  o  .H     C.iaia^^ii^a*^^ 
AH«   o     «h-3PH       pqpq^pu:   WW^MN   ooS^M   £ 


*$•  vo  00  ON 

OOOvovo       M    M    T|-  co  c*     vo  oo    co  O    co 


S 


ON  VO  M          t^ 

vOOOVOe«<<*-vOOOOOOOOOOO<<*-c<'4-f<  Of»ONONCO___ 

t^.cOHiTj-'^-      HMH^-ON  MVOM    ONVO  t^-vocovoco  MONC4OOO       ^-  rj-oo    rt  OO       rf-  ON  CO  M  vo 

ONCOOOOHi       VOMCOMt^  HOONrJ-T*-  voONOOOOO  ONOO    ^-   H     M       VO     O     t*»  t^vo        ON  M     t^-OO 

co  co  ONVO    r<       ^J-vO  oo    covo  ^-  co  co  r^  T*-  co  O    ONOO    c4  VOCOHVOCO      coclONMrt       ONOO    O 

oor»MTj-Tj-      Ht^c4    covo  OOOOONrJ-  OOoorlcJ  IOONVOVOVT 

OOONt^-COCO        ONONVOCOt-^  ONONOVOl-^  t^VO     OOOt~-  ONONQMOO 


M  00  00  VO        •<*• 
ON  t^-OO     ON  M       OO     O  vO    vo  vo 


ONONONONO       ONONONONON      ONONQONON      ONONONONON      ONONOOONOOONONONO       ONQONONON 


00  M      VO 


vg 


t^  vo        ^-  ONVO      ON  Tj-  M  OO      ON         ON 

VO  ON       VO  ONOO 

q  q  q  o  o  o  q  q  o  ON  q  q  q  ON  q  q  oo  ON  t^-  q 

M*  M'   M"  M*  M*     M'   M'   M*   M*   O  H."  M*   M*   O*   M'  M*   O  O*   O   M     Hi"  Q   Hi"  M"  M"     Q   «"   M'   M" 


ON 

8rt    O    rt  ON 

f^OOO       ONOOQvo     OOOooO 


ON   O     CO  O     HI 


vo  O    M    vo     ONOO    M  vo    t~-      f*    O  vo 


co     M    M  vo  oo   co     vo  t^oo 


VO    t^-00      M      VO 


ON  c* 

Tj-     M 


HI  CO    CO    M 


^VOMOO        t--r-sMOvo 
t>»t^coo      vovoclooN 

M      M      HI  H      M      M 


T}-   rj-  ^   vo 

1VO    VO      CO    ON 

d         co  M 


ON  O    vo  t^OO 
vo        c4 

VO   H     M     CO   CO 


$ 


VOOO     Ht^-CO       COON^-T^-IO       COM     Tj-VO   OO         rj-    O     vo  t^   t^ 
MfxdvOO        VOCOQOOM        rJ-vO'tj-voO        OMt-^MOO 


rHrH      C<1  r-l  Cq          rH 


Illil 


rj  "HI     •  bo  t>    *S   rj  -S  bb  -S 
"^    Q^lrJ    r-s    o       CL.^    A  3    5 


SO        PH-O; 


C<J  CO  CO  rH  t-     00  OO  t>. 


00 
1C 


OS  O  O     rH  <N  CO  Tf  OS 


-£    >    >>  >    > 

$&%&& 

O  O  rH  r-  1  CO 

oo  oo  oo  oo  oo 


I»  l>.  IN.  i»  i>.     i>«i>.  c>»  !>•  QO    QCaDCCQOQID    OO  00  00  QO  Ct 


640 


otctot 


ooooo 


TABLE  XV1IL 

Elements  of  the  Orbits  of  Comets  which  have  been  observed. 


"a 
1 

.S        .S 

*-§'!  s'l 

Encke. 
Gibers. 
// 
// 
Burckhardt. 

Wahl. 
// 
Burckhardt. 
Gibers. 
Gauss. 

c5  I'- 
ll 1|1 

Bessel. 
Triesnecker. 
Argelander. 
Nicolai. 
Encke. 

"H 

1  fa*  i 

Hill 

Motioo. 

I  1- 

r£    'c\r<a\'c\ 

Retrograde. 
// 
// 
// 
Direct. 

// 
Retrograde. 
// 
Direct. 
Retrograde. 

S 

// 
// 
Retrograde. 
Direct. 
Retrograde. 

it 
Direct. 
Retrograde. 
Direct. 
// 

Retrograde. 
// 
Direct. 
// 
// 

I 

Tj-oo  vo  vo    O 
M    co  r^-  M    vo 
r-x  vo  c)   vovo 

c*  vo    ON  covo 

vo  O  oo  oo    O 
ON  O*    ON  ON  O* 

M  VO    O  VO    rj- 
OO     vo   vo   CO   t-*-. 
ON  T}-  co  l>-vO 

M      M      VO    VO    "4- 

O    w  oo    O    t>» 

ON    M      ONVO      M 

ON  O*    ON  ON  O* 

vo 
^  i    ON  O    ON 
O    vooo    r-^  r> 

CO    M      T^-     COOO 
Tj-00      M      VO    M 

H    ON  H  OO    ON 

VO    *H      t^x  VO    OO 

ON  O*    ON  ON  ON 

t~s    CO    »^-    M    OO 

cooo    O  vo    vo 
**•  r}-oo    O  oo 
•^}-  vo  r-x  ON  ON 
H    ON  M    co  H 
ON  f^  rj-  O    O 
ON  ON  ON  O    o" 

00     O           00 
VO     Tj-    ON   vo 

O  vo     M     CO  ON 

H   r--  TJ-  o   o 

CO   VO   CO   >H     ON 

vo  a\  o  oo   vo 
ON  ON  O*   ON  ON 

oo    ON  vo 
O    vo  t^  vo  ON 
t^  vo  M    co  ON 

OO     CO   M     C»     ^- 

CO    ON    VO    ON    O 

OO  oo    M    ON  ON 
t~~  ON  O    M  OO 
ON  ON  o'   O'   ON 

ON  t»    ON 
l-^  M    O    ON  O 
vo  c»    M  vo  VO 
vo  ONOO    t~-  t» 
r}-  ^  co  10  vo 
•<j-00   00   00  VO 

oo  q  o  vo  oo 

ON   O'     O'  00*     ON 

• 

00000 

5- 

CO 

0    0    0    0    ON 

OO 

OO 

00 

oo    O    O    O    O 

00000 

ON               M 
H  00         00 

t^  O  oo  oo 

M       t^-     1-1       Tj- 

-4-  ^-  M    ON 
oo    r^  O    ON  O 

O  OO 

cooo    f» 

co  o    M 

O    t^  vo 
vo  H    -*• 

ONOO    vo 
O    O    ON  ON  ON 

00 

VO 

ON 
M 

0    0    ON*  0    0 

° 

° 

M      M      O      •*      <-> 

^   M  o  ^*  r^»  to 

t-  l^rh  0    0 

0    co^-c*   * 

«    *°    £3- 

O    •<*•  cooo    f* 

ON  vo  M    M    to 

CO   C<     VOVO     Tt" 

s. 

"*"       M      H      CO    Tj-    VO 

d      VO   t^    W      M 
VO    Tj-                C<       tO 

t>    rj-  O     'J-VO 

rt    vo  O    O  oo 

co-1o"2? 

OO    M    cJ     t^  C^ 

MM                 M      VO 

co  rt    ON  vo  H 

°     °3~    M    VO      t^^ 

VO    co  -<J-VO    vo 

M  VO     vo  -4-  Tj- 

VO   t^    t»     >O  VO 

M      M      COVO      rj- 

tovo    r-to^ 

c*  oo  ^  •**•  tJ 

^     vo  covo    c^    t^ 

CO  ^-    rt              CO 

^     M  vo    rj-  O  00 

«     ON^-Ot, 
M     M     •*    ON   O 

d  VO     t~x   M     O 

H    w    co  rt    co 

cooo    O    ONOO 

M              CO  VO 

0      ONVO      M    VO 

iT  rt  ^  Jo  ** 

•4-  vo  T}-VO    O 

f»     M     CO  VO 

CS 

VO  VO   H     VO 

VO  VO    H    t»    t^ 
O     o    vo  vo  t-^vo 

to  O    cooo    c< 

CO   ONOO     O 
M      Cl      1-1 

^J-  t~»  ON  rt    ON 

CO    M      c<      C^      ^" 

to         co  M    tJ 

ONVO    Tj-  O  vo 
ON  r>    'i-  M    f- 

CO            CO   M 

Tj-    M      f*    VO      t< 

co  vo  r*  vo    r) 
to  6)    co  rJ    co 

^h  O    O    to  to 

H     HH     rj-   ON  vo 
CO   M              C4 

O    c*    co  co  O 

VO     TJ-00     H     vo 

It 

ft     ON  ts.  Tj-  vo  1-. 
VO  Tj-    CO 

T£OO     ON  Tj-  rj- 

t««  ON  ci  oo    w 
o         ON  rt   vo  M 

t^  cl    vo  O    to 

C<     CO  CO 

co  O    ?<    f*    rj- 
•«h  t*   vo  Tt-  vo 

COVO    VOOO    11 

rt"  ^  nT  c*  "^ 

8    cooo   t^  l-^ 
M               VO   tl 

VO     H     ON  VO  rj" 
vo  ON  T|-  O    to 

VO  vo    O    -^-  w 
00    rt    ON  ON  r»- 

CO  rl     rj-           T}- 

to  O    co  ^  oo 
ONOO    co  rj- 

M      M      CO    M 

t-^00    co  rt-  t* 
•<*-  H          10  M 

vo    ON  t-*  O    ON 
«o  O    ON  t^vo 

VO  T»-    CO   rt     «4- 

oo   ^  O    i~^oo 

CO  ^"            H     ^ 

vo  vo  r-s  r}-  ON 

00  00  VO     CO  O 
VO   CO 

vo    to  M   to  t^ 
vo  ^J-         co 

ON  t-~  ON  f-  vo 
vo    ON  Tj-vo    ON 

*>  ONVO    ON  ON  ON 

CO   C»     CO 

S  00     vo   VOOO   VO 

rj-  vo  ON  O     1 

VO    M                           H 

vo    o    r^*  rt  vo 

n    H    O  vo    co 
vo  t~»  w  OO    f^ 

vo  vo    O    vo  vo 
CO  rj-                   VO 

O    to  co  covo 

O    O    H    ON  rj- 

10   M               VO 

ON  M    ON  co  r* 

VO   CO   CO   M     CO 

O    to  O  0    M 

O    1--  d    O    O 

oo    M  oo  oo   vo 

<  ON  t~»  f>.OO    f^ 

vo  t»    l^  O    w> 

O    ON  t»    M    co 

VOOO      CO    M      TJ- 

H,      f*      f»      t-    t> 

5j-  rt  VO    co  t^ 

t»    O    coao    O 

^  f^\  &  l~5  ^ 
CO  00  00  OS  OS 

CO  rH  CO         CO 

§  CO  CO  CO  CO 
OS  OS  OS  OS 

CO                     CO 

io  o  t^  co  co 

OS  OS  OS  OS  OS 

iMslM 

OS  OS  O  O  O 
t~-  t^  CO  CO  OO 

jllft 

CO  OO  CO  00  OO 

S  co  SSi5 

00  CO  CO  OO  OO 

lllll 

CO  CO  lO  CO  CO 

oo  oo  co  no  co 

OOOO'H 

i-l  d  W  *#  i« 

2£22i 

SUSSw 

gJSSSw 

1-1  SI  95  rfl  »» 

CO  CO  CO  CO  CO 

09  WCO  CO-* 

iHlHrHlHiH 

41 

641 

TABLE  XVIIL 

Elements  of  the  Orbits  of  Comets  which  have  been  observed. 


" 


ver. 
geland 
mbart. 
Heilig 


1=  £rs^      -g  5  s^  £ 
o-3O  >      owo^w 


R 


ONOO     vo 


f-oo    O 


co  co  cooo    to  vo    ON  to  O    "<$• 

ooootodoo  OddONt-~ 

r~«  d    d    cooo  tovo    O    co  d 

O    ON  to  tooo  ON  ON  t-~-  to  ON 

O"    ON  ON  ON  ON  ONOO'    ON  ON  O*N 


tovo    l~-  ONVO       *3-oo    O    d    M  to         d 

OdQvOi-i       d^i-oooooo  t^  d    M 

co  r~-oo    Tj-vo       ON  co  w    o    to  d    tooo 

OO    O     t^»  d     ON      M  vO    t^  Tj"  vo  T$-  OO    tr 

OQtOMMOO        VO     t-^    CO   to   M  ^-    O      O 

^-  co  ON  to  O 

ON  to  O    O\  co 

O'    ON  ON  O'   ON      ON  ON  O>'   ON  O*  ON  ONOO*    ON  ON 


t-.00     d     d    to  t-  «0          rj-  M 

rj-      to  co  *J-  d    O  O    f--vO    O    ON 
ONOOOvDOQONVOdro  covo 

«$•       CO  to  rj-  ONOO  1-1    M  OO  OO     O 

t^      ONOO    **•  ON  to  co  ^vo    M    d 

**.    °^  ^t"  ^  ^      M.    *"?  °^  °.    *?  °.    °^^D.    ^  *** 

ONO>ONONON  o'oNoVoNO* 


to  M 

M    co  oo      ro  M 

'i-   O  «o  rj-  d 

00     ON  rj-  VO  O 

to    M  t^  T*-  co 

OO     to  VO  TJ-  VO 

rl-     to  OO  TJ-  ON 

OOooOt-^voOOooO  ONOO 


oo  oo    M    ON                                      co  rj-  T^- 

oo    d    O    to                                    t-~-  d  M 

oo    rj-vo    ON                                      d  vo  •^« 

TJ-  tovo  oo                                        ON  r*-  to 

TJ-  ON  rj-  O                                          ON  ^  rj- 

OOOONt--O  OOOOO        ONOO    O  O  00 


MMOMO   OMMOU   OMMMM   MOOOM   MMMMM   O  O  «  M  O   MQMMM 


CO    W 

u-i  o 


O    t-~  ON  ONVO       t~^  •^* 
M   to        to  ^J-  d 

ON    M      T$-VO      M  MM 


•**•  t^  o  c 

CO  d    co 

Ovotod     ooMoovod 

d      M      T^"    d  M      M      M      tO 


Tj-vO  t^-VO    rj-  ON  co  d    O    ON  ONOO 

TJ-  rj-  to  «o  d  ^*-  co 

coO  co  O    to  to  M  rj-coo\  tovo     NO 

T^-     M  M      Tt~    d      d      tO  ^"  M      M  tO 


to  ^J-OO  ON  rJ-VO  f^ 
cooo  M  rj-  co  O  co 
rl  M  co  M  to  d  co 


MdCOl-IT^-         MtOCOw 


tOt^-ONtOO          OOONt-^MCO         O      Tj-VO      M      «O         Th    TJ-VO    OO      M 

d  covo  r~»*J-  t^ONdt^ON  MTj-o^hco  O  r^oo  ON  t^ 


to  t~>  to  co  t-x 

co  to  M  co  ON 

co   d 


Ot^-dOt-^  f^ONTj-vovo 
totoMMto  dOdt~^O 
d  HI  d  co  M  d  M  d  d  d 


Soooodvo       cod 


M      co  to 


T*-    oo  v 
d       co 


M    co     r^-vo  vo    O    co 


eoscTi-rcrcr  CO"«O>ONOO*V^" 

(N          i— I  (M  CO      i-lrHr-(r-<CNl 


w  X  zs  BJ  ^  oi  .S;  k_j  ~r 

OQt-302^  ^OQQ^^ 

OOOOO3OiO5      O5  r— I  GNI  CNJ  CM      (MCO^^lO  lOtO>-O?OCD 

rl  r-l  ^  Jli  ^H      ^-t  CM  CNJ  (M  CN      (M  CN  ^  CM  C^  (N  CN  CM  CM  CM 

cooooocooo    oooooocooo    ooooooooco  c-oooooooco 


^  ^¥^1 

^_  ^ 

>*   -*4  >  •"  >!H  ^ 

°?    s^  o 


CO  CO      CO  CO  CO  CO  CO 
COOOOOOOOO      OOOOOOOOOO      OOOOOOCO'CO 


642 


TABLE  XVIII, 

Elements  of  the  Orbits  of  Comets  which  have  been  observed. 


Encke. 
Westphalen. 
Encke. 
Peters  and  O.  S 
Plantarnour. 


g.S 


Dir 
ii 
n 
Re 
Di 


Direct. 
Retrograde. 
// 
Direct. 


ONTj-ioj^coooOHivotovodooTj-vo  «*    O        ooo 

OO    ONOO    n  vo  -^h  t---vo    Qvo        M    co  O    *j-  d  vococoQOO 

O    HI    o    O    to  ON  O    co\O    r-~      covo    covo    HI  vo    co  d    O  vo 

M    covo    cooo  O    vooo  voco     to^-dnON  r-.  tooo    M    T^- 

t^oo  vo    M  o  rf-Ot---dcoooooTt-dO  voooroONO 

covo    co  ONOO  l~x  r^  co  O    «3-      O    d    l--  co  o  vo  ON  O    d    t-~ 

tor~.toi>.q  oo    M    to  t^.  t--.     d    d    O    ON  •*  O^  O  vo    to  M 

o\  6\  o\  o\  d  o\  o*  o\  O°N  t>.  o  o"  o  o\  o\  o\  o  o\  o\  o' 


»H    ON  co 


ON  d    M    d    ON  toodnto 

OOOOOOON  oo  vo    O  vo    o 

t^  ON  ON  d    d  oo    c~^  co  r***  o  vo  d 

ddnoorj-  wOOONiOt^.  MOO 

co  «  d  cooo  o  M  d  d  -*•  r^oo 

OjvOO  oo  HI  HI  oo  ONVO  co  d  M  vO 

ON  ON  ON  O  O*  ON  O\oo'  O  O  O  ON  ON  o"s  ON 


t^ 
vo 
r^  ^-  r^ 
w  o  rt 


vo  ON  to  O  vo  vo  T*- 
vo  O  l-^  to  co  d  O 
CO  ON  r^  O  OO  to  ON 
O  co  M  d  oo  oo  t^ 
to  r^.  to  O  r^  ON  T}- 
TJ-VO  Tj-  O  ON  vo  ^J- 

oo  ONOO  O  ON  O  ONOO  O 

O*  O  O  HI*  o*  M  O*  O*  HI" 


t^  oo  t--.  ON  co  O 

vo  ON  ON  cooo 

HI  r^  ON  to  O 

ON  HI  oo  vo  vo 

ON  O  vo  t-^  ON 

ON  O  to  HI  ON 

ON  O  vovo  ON 


co 


d  d  vo  O  O  w 

rj-vo  d  vo  oo  ON 
t>-  co  o  O  oo  oo 

oo  •>*•  T!-  vo  co  o 

ON  t-^  d  vo  co  d 

OO  Tj-  ON  vo  ONVO 
O  O  ONOO  ON  t~-  t^  ON  O 


00 


to  ON  t^  ON 

X3  oo  d  d 

CO  CO  M   HI 

co  ON  co  ON 

HI  ON  co  ON 

d  OO  ON  ON    ON   ON  t^ 

t^  ON  ON  0-.  O  ON   ONO> 


0         vo 
rj-  O    H 

COVO     CO 
-J-   U1  H, 


O       HiOOOni       nHiQOO       OOO>iO       OOOHiQ       OOMHiO 


vo  vooo    d    O       d    covo    t^  ON    vo    co  O    M    t^     ONVO    ON 
HI         Hivod       voddcoco     rj-covo        T}-     cocovo 


^        M      to    M      IO    CO         Hlt^. 


S    c^g-      $ 


d       MTj-tocod      wvocodto 


n    10  to  to 


a 


•*  vo  Tj-   ONVO       VO  00 
O      COlOCOHi     CO     OO^t- 

co        co  M    d       HI    d 


VOQVO     COM          CO»^-COCOM 

Hid  HI      cocococOHt 


•4-  10 

U-,   w 


CO       ^    M 


dcodHiM       Hicod 

vo  o    vo  ONOO      d    d   to  d    t^ 

HICOMM  CO  MCOd 


M    ON  d    ONVO       HI    d    d 
00    •*  ^-  t-^  ON      ON  ONVO 


dMoodd      t^vooNONON     covo  vo   to  d      ON  r>- 

CO  COrJ-COdVO  CO  CO^-COHITJ-HICO 


ONMVO        OOOVO^-VO        OONO>MO 

co  o  vo   to  cf 


^*i 

_OO      OOC^C^CO      COCO^Tflrtl     lOiO>OvOO      COCO«OCO?O     COCOr^t~-t->      h'.t^.t^-OOQO 

OO  OO  CO   OO  CO  CO  CO  GO   0000000000   GO  00  00  OO  CO   CO  00  00  00  00   CO  CO  CO  00  QO   00  00  CO  00  00 


v*G*C6'*#*6    S3t- 

GCQCaDODOO     QCQCQCQOO 


?Df>.Gt  CtO 


QCQCQCQCiO     C&CtCtC^Ci     C6  d  Gi  Ct  O 
«-,-  —  —,—     ^_  __  —  —  —     fHfHrHTHOl 


ooooo 


TABLE  XYIII. 

Elements  of  the  Orbits  of  Comets  which  have  been  observed. 


*  s  §  § 

•S|  iiia-s  *:s 

3§    ££w2w 


53    00  •       .  g    J  O>     03     £ 

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b^^rCO      L-3  ^4  £2  »X  K*i      h"!  7-5^  O"!  h>  ^      ^^i^T-S^r^M 


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0  0 

H   vovo 


ON     w    o 
Th    oo 


f-^OO     vo 


O   •<*•  to  6  c5 

T*-  c*  vo   c»  i- 

O    HI    ONOO  HI        cocOt>vot>      OO    vo  rj-  t>-v 

rf-OO  OO     HI  Tfr-       IH     rt  OO     vo  HI        to  t-~  HI     ON 


ON 
O 


6.  ..     ..     rt     t>.  O 

H  «*•    •<*•     HI      M      H 

CJ     ^"    HI     rj-   rt  O     ON  CO  cJ   OO         O     "^  VO   OO   OO         r^   t^OclHl         C)C)HIHH     t^^       O     r^   t^     vo  CO 

OO  VO     vo  CO  vo  tOVO     ONVOt*       VOtOONtOvooOCOi-iTj-i-i        OcOr}-votJ        ONOO     ONVO     t~^ 

ONOONOt^-  rJOONHivo     O\O\OOON     •^-t)to  rj-oo       ONi-<cot-~vo     Ooot^  vooo 


tOOQ     O     vo 

O  ^O    M  oo 
O    O    O  oo 


ONOONOOx     OOONONOx      ONONOOON      ONONOONON      ONOOONON      OONONONON      ONOOOON 


co        ON 

VO      O      M 

OO    co  vo 

oo  oo  oo 
t-~  r-^oo 
ON  ON  ON 


O    t^oo 

VO  VO  10 
ON  t>  00 
Tj-  ONVO 

_._._.  VOVOON  ,  

O    0\  ON  ON  O       vovo    ON  O  OO        O    t>-  ON  ON  ON 
HI*   O*   O"   O*   HI'      o    o'    o'   w    o*       H*    o'    O'    O'    O* 


vo  HI  O  ON  ^t-  vo  ON 

H  ^i"  vovo  cJ  ON  rj- oo 

r^  O  vo  HI  M  w  ti  cl 

vo  vooo  ON  TJ-  to  TJ-  H 

r--  H  VOOO  O  ON  ON  HI 

vo  vo  w  ONOO  H  O 


VO    M    O    O    ON  T»-VO    i*-  -4-  vooo    •<$-        vo 

Tj-  T^-  vo  r^-vo  vo  T}-  rt-oo    HI  to  «    ON  co  co 

M     O  OO     ONOO  vowONONr-  HiONOOO 

cOTt-wONt^  HHHONCO  ON  ONOO    ON  >-i 

CO£>votOC^  t^ONrtOOO  VDVOONO'i- 

VO     O     •*  ON  ON  O     ONOO  ON  ON  vo  M     vo 


O  O  O  O   ON  0\  Ov  ONOO   ON  OsOO  ON  ON   ON  C^vO  OO 
M  M  w*  «   O*  O*  O*  O'  O*   O*  0  O*  O*  0*   O*  O*  O*  O*  O* 


tO  O  ON  rt-  CO  00  00 


00    ON  O    O 


vo     HI 


HI    coox)    coco      ONHOOt^      HiOvOc^Hi       O^-HICOCO     Ol~^  ONOO    t»     vo    t-~-  co  rj-  o 

M     M     tO   f^    HI         ^*    M     T^    H     VO      VO   VO   VO    OO     t^        ^    M     VO   61     HI          hH    OO     tJ     vo   CO       VO   CO   •-(     vo   IM 


HI        OOOOVOf-.       t<c»'i-t--.vo       HI^-  COOO    ^       HI     t~^  VOOO  VO       OO     d     t^  CO  ON 
C>          COC4  COCO       Clvo^cJ^J-        COVOCO^J-H          COCOwHHI         TJ-Tl-  «vo 

t>^  co  O 


ON  t~«      ON  to 


co  in 

O  vo    covo    r^oo 
c* 


co  ON      ON  rt    O  00     . 

^•f^HICOVO          t^O't'VOO  HI 

COCOCOM       C<HI          HIH       c^ 


OOONCOCOM          OHVOCO 

°     :          :       co  co 


Tj-    VOVO      ONt^       VOTt-VOONHI  O^htOVO 

ONVO    t<tovooot^HiTt-cJ       VOTJ-HI-I 

HI     rJ  C*  CO   w 


ON      rh  H,    ON  O    HI 


CO  HI   00   00 
«     W  VO 

H    t^  HI    0 


Ill 


OSO5O5OO  rH  i— (  i— I  i-<  <M  (M(M(M 
rtl-^^tOtO  tOtOtOtOtO  tOtOtO 
CO  CO  00  00  CO  CO  GO  CO  00  00  CO  CO  CO 


CO  CO  CO  rfi  -^  rtl  -^  rf  to  to  to 
tO  to  to  to  to  to  to  to  to  to  to 
CO  CO  00  00  CO  OO  CO  CO  GO  GO  00  CO  CO  00  00  00  CO  00  GO  CO 


t^  t^  t^  GO  OO 

tO  to  to  to  to  to  to  to  to  to 
0  GO  GO  CO 


,-1^1-11-11-1  THTHT-|!-H<N  09  09  91 91  SI 
<N  (N  «1(51*1 


rH  (N  SO  -^  1(5 

to  w  w  «  cc 


644 


TABLE  XVIII. 

Elements  of  the  Orbits  of  Comets  which  have  been  observed. 


a 


41   a 

«?3    *53 


Hertzsprung. 
Liais. 
Seeling. 
Liais. 


83  -is      Si 

:|  1*   .   .5  fid** 

O      'J3    2^$vg    g  ^^.§,§,0 

•§    S  §g  ST3  o  gl'-S-g 

fa   M£^P3>  M  g£i>H 


fe 


Ret 


Direct. 
Retrog 


O 

vo 


Tj-  •<*•    ON  VO                       d 

O  co  co  TJ-  vo  ONVO  O 

r^OooOd  d  vo  M  o  t-^ 

vo  oo  d  rj-  M  ^t-ddt^vo 

d  vo  d  d  ON  TJ-  cooo  vo  vo 

oo  covo  co  d  vo  O  f--  w  vo 

O  t^-  t*^  vo  d  M  co  o  w  ^J* 

o"  c?\  o\  o\  o"  o°  oV  o"  o*  o\ 


i    O         VO  oo          ON  r-N 

oo    ^~  cooo  oo    T^  vo  ^ vo 

M    t^%  M    ^-  M  vo    t-^  co  O 

vo  M    Ooo    co  oo    •^-•<4-dvo 

ONTJ-VOCOM  M     COTJ-OOO 

t^vo    i-i    d    co  ONOO    O    O    d 

OvONONOjNVO  ONOJNO\O\O 

ON  ON  ON  ON  ON  ON  ON  ON  ON  O 


vo  M  T»-  •<!-  H 
VO  t-v  TJ-  O  vo 
d  n  co  co  M 
oo  ON  r^oo  d 
ON  -3-00  M  d 
t~-.oo  oo  M  oo 
ON  ON  ON  O*  ON 


O  O  d  vo  r» 
O  Thoo  co  o 
t-^  O  ON  n  d 

OO  ONVO  f-.  vn 
VOVO  OO  **•  w 

ON  ONOO  O  ^J- 
ON  ON  O\  O  00 


CO  t--.  ON 
n  O  vo  O 
oo  f^oo  O 
VO  OO  vo  ON 
ON  vovO  O 
00  CJ  ON  vo 
ON  d  M  O 
ON  0*  0*  0* 


VO 

d    vo 

CO    M     V 

ON  ON 


346 

538 


ON 

ON 


Met 


O    O    ONOO 
M'    M'    O    O' 


•00000 
O       M'    M'    M'    M' 


OONO\Ooo       OOvOONO       OOOOO 
M    o"    O'    «    o       w    O*    M    o'    «       M    M    w    M'   M' 


t^  ON 

vo  ON 
ON  ON 
ON  ON  O 


O    O 

0*    M    M    M 


0.9054198 
5575382 
.8490551 
O 


0 
0 
I 


d    t^OO    "*•  M 


co     M    d    w 


S£? 


t^  d  vo    co 
M    co  w    co 


ON  f-  -^00    rj- 
vo  vo  d        oo 

M     CO   CO 


HI  d    n    CO      CO  M    r*-.  M 


VO   M     CO   O     CO       M     ONOO    OO 

ON  co  O  vo    vo      co  O    f^vo 
d    «    d       d    d         M 


vovo  vo    t^  ON 
O     ON  d    co  vo  Tf 
M    d          HI 


O       VOONdTj-TJ-       QHIOOHIM         OHIOO 


1-^00     vo  vo 

•*  ^-  M   d 

ON  ON 

d    co 

O\     co  o\  HI    r«*oo      co  O    O    O 

HI  M  M      HI  d      d 


C?iQO?OCC<? 

<N  r-t  i— I      i-| 


co'io'co""  o6~co"r-T. 

r-(        r-i     (M        i— I 


vO  rH 


1  Oi  O  O  O     O  r- 1  i-l  i— I  <M     <M  <M  OQ  CO  CO     CO' 

lOCOCOCO      COOCOCOCO      COCOCOCOCO      CO1 
COGOOOOO      OOOOOOOOCO      COCOCOCOCO      OO  i 


i  rt<  ^  iO     CO  CO 
'  OO  OO  CO      OO  OO 


(N  (N  «!  »1  SI 


645 


TABLE  XIX 

Elements  of  the  Orbits  of  the  Minor  Planets. 


'  Illl! 


a.  0  o  •<  <3 


tO  iO  *O  iO         i£N 


co      cc  co  co  oo  co      co  co  co  cc 


O         "^  co  d  t^  O   co  t*x  d  co  ON        T$*  t*x  O  co 

ON  vo  d  O  NO  oo   covo  VOON  co  oo   T}-  T$-  ON  OOcovod  oo  oo  t~~-  d   O 

vo  M  «o  d   ••i-  vo   M   ONOO  vo  Ocovocoo  codO   ONVO  t^  ON  co  vovo 

ON  Tf-  d   d   M  oo   t^vo   t^-rl-  vo  M   o  vo   d  ONCOONVOON  VOONVOCOCO 

M   d  vo    co  M  Tl-t^d   r~-oo  ONOO   Mdd  vo  co  o   t-^  M  vo   co  ON  r^  O 

rl-rt-dt^M  OO    f>-  TJ-  f^  ON  OOvOMwd  VO    ONVO  OO  OO  OO  VO    M    ONOO 

TJ-rj-Tj-coTi-  cp  co  cp  cp  T}-  rprr|'*"'^"':J"  rj-  co  cp  cp  co  cPT^"T}"Ti"rP 

66666  66666  66666  66666  66666 


ON      f^  r^-      co 

ON  CO  vo  -rt-  O  OOOMVOOO 

covo   OMd  oo   •*!-  d  vo   ON 

d   rj-oo  d  ON  vo  oo  M  >H  ON 

rJ-Ocor^co  co   d   r~-  ON  r>» 

d   t^  •*}-  O   t-»  ON  M   vo  d   t-~ 

^J"  cp  TJ-  TJ-  cp  rJ"T*"Tt"rt"Tl" 

66666  66666 


VO 

oo  vo        vo  O 

M  VD  OO  OO  d 
T}-  ONVO  CO  vo 
d  O  O  coo 

q  oo  q  vo  vo 

M     ONTJ-      " 


vooooooo  vodMTl- 

d  vo   ON  ONOO  vo   t^-vo    d   O 

d   O  O  oo  M  co  T$-  ONOO  vo 

CO  OO     CO  CO  II  VO  CO  r*.*  O    VO 

Ovococoro  M  oo  oo  d  ^h 


co          oo  oot~»       vor~>oooooo  rj-oo  t^ 

covo  oo  t^.  r^  cooo  r>--«i-d      vorj-d^j-co  OOONONCOM 

O     VO  ONOO  OO  rl-  COOO     COCO         ^-OdTj-Tj-  OVOVOVOTJ- 

vo  vo  **  t^  vo  vo  M  o  vo  d      oo   ON  d  co  t*^  vo  t^*co  vooo 

r^*  OMd  vo  vo  vo  M  f^oo      vo  ON  M  co  d  oo  vovo  oo  t^ 


M   l-^.  vo 


ON  N  vo   d   -4- 

COVO  OO  VO    CO 

OsONO   ONVO 


T$-  T}- 

c>   ON 


CO  VO 

vo  vo  d 

ON  ONOO  OO  OO 


ONd  O  O  oo 

O     w     d     COrJ- 

t^  ONO   ON  ON 


CO  rj-  COVO     CO 
CO  11    CO  CO  VO 

ON  f~-oo  vo   ON 


ONVO  VOONVO      codMVoO 

MCOVOVOt^         COVOCOQOO 
OO    ON  t^OO    O>      VO  OO    t^-OO  VO 


OO    VOOO     COM         OOCNjCOrl-Tl-         MONM 

d  t^  vovo  6       4-  O  vo  d  vo 


M    d     CO  CO 


CO  IO  VO 
OO 


O  d  d  d  co      4-oo  oo  d  06 

M     d     COM  ^4-  M  VOVO 


VO  CO  CO         CO  I 


COOO     Tj-  M 


Q 


co  rj- vo  O       M  co  ON  t-^  vo      «o  d  Tj- ON  O 


i  O   O        ON  O   O   co  d 


d  d  ONOO  vo 


vo  T}-VO   M   cp 
ON  l^-vo  d  vo 


t--OO     CO  CO  O\ 

^cT^d  5^"° 


q  q  <?NONVO 

vo  vo  ONOO   t~» 


t>>  vo      TJ- vo  vo  vo  co 


t^  co  O   Is*  "*J" 
co  d  co       ^« 

•<1-00  vo    ONM 


CO  vo  O   M   O 


VOVOM 


d  vo  oo  M  co      cooo  vo  d  oo 
t^»  vovo    d  OO         t^  vooo    vo  w 

vocodM       Mvorhco 


I  -4-       t~~  •<*•  ON  ONVO         t~-  ON  t^  d 


cp  vo  Cfr  d   d 

ON  M    ON  »O  ON 


ON  vo  ON  vooo 


.  vo  O   d  vo      r^oo  M  O  ON      M  oo  vooo  M 


O   vo  O   w  vo 
vo  d   vo  M   O 


O  vo  r^vo  •<*• 

00  vo  VO    CO  w 

d 


VO  CO  rj-VO  OO 

rl-  ON  rt-  vo  O 

M    CO  CO 


M    O     ONTj-VO 

co  d       oo  vo 
d        M   co 


tJ-OO    ONVO    w 
00  VO    M  00  00 

M  cod   -*• 


•i-  q  q  oo  vo 

M     VO  VOVO     6 

co  d   M 


vocoti-coO        T}-ONVOI 


d  w  co  vo  co 

vo   rf-  vo  t~~-  ON 


rj-  M  VO    VO  Tj- 
00  vo    M  vd    CO 


OVOVOOOON      vOMt^dO 


to  vo  ONOO   ON 


OO    d    rj-  ON  vo 


VO  M     N     M     VO 
W    TJ-  CO  t^  CO 

H 


vo  O  vo  O  oo 

M  VO     n     CO  ON 

d 


t^OO   coO   d 
d  vod  rj-0 

tO  M     IH     CO 


vo  t^  d  vo  M 
coco  d  vo  co 
d  M 


CO  r}-  d    O    M 
ON  ON  T*-  voO 

M     CO  M     d 


t^  O  ONOO  vo 
VOO*NOO  vod 
co  TJ-  d  co 


_      t^  M     ONVO     Tj- 

O    m  M   d   M   co 


vo  q  vo  t^  q 
6  6*  co  4-  d 


H  VO  11 


MONCpclvo  vOMQcpd        MO_NCO  t^oo      vo   cooo   cp  O        O   O 

vo  ON  M   ON  M  vo   4"  l"^  co  4"      vovo  vo   O   M       vo  M   r^  d  vo      d   4" 

llCOrl-  CO  T}-dCO          rj- 

•rh  d  T}- vo  vo  O  t»»  r}-  vo  co 


=  1   £e-* 


il  -sf.pl  HI'L 

1    3        m'j    ®0^       "S'djajJ 
ifq      PL|HS^S      H^OHHPn 


lMl.2  Ii 


II  Is 

3    0    S*    tn 


P-  S  ^>  «    3 
3   c   o  .- 


Ol^0004O          r* 


646 


J.  xi  JJ J_LU     WiYV i 

Elements  of  the  Orbits  of  the  Minor  Planets. 


i    1: 1 1  I       aa         a  a  d     .  a     a  a       I  *i  •.£  d  £  a 

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2-5§§2  2  Softs 2  So:222£?  £25213  2:2:2  S  **  Seg-gg  SS)5|2 


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nJOi-^Oi 


CO    T-l 

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t-        CO  iO  OS  OS  T* 


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TABLE  XXI,    Constants,  &c, 


log 

of  Naperian  logarithms e  —  2.71828183  0.43429448 

Modulus  of  the  common  logarithms         .        .  A0  =  0.43429448  9.63778431  — 10 

Radius  of  a  Circle  in  seconds r  =  206264.806  5.31442513 

'/         '/        a       //  minutes r  =  3437.7468  3.53627388 

"        »        //       //  degrees r  =  57.29578  1.75812263 

Circumference  of  a  Circle  in  seconds        ....        1296000  6.11260500 

"  //        //      whenr=l.        .        .        .   TT  =  3.14159265  0.49714987 

Sine  of  1  second 0.000004848137  4.68557487 

Equatorial  horizontal  parallax  of  the  sun,  according  to 

Encke '  8".57116  0.9330396 

Length  of  the  sidereal  year,  according  to  Hansen  and 

Olufsen 365.2563582  days    2.56259778 

Length  of  the  tropical  year,  according  to  Hansen  and 

Olufsen 365.2422008    //       2.56258095 

This  value  of  the  length  of  the  tropical  year  is  for  1850.0.      The  annual  variation   is 
—  0/0000000624. 

Time  occupied  by  the  passage  of  light  over  a  distance 
equal  to  the  mean  distance  of  the  earth  from  the 
sun,  according  to  Struve 497/827  2.6970785 

Attractive  force  of  the  sun,  according  to  Gauss         .        k  —  0.017202099          8.23558144  — 10 
//            //        //         //           //         //      //  •     in  se- 
conds of  arc 3548.18761          3.55000657 

Constant  of  Aberration,  according  to  Struve 20".4451 

//         a  Nutation,  //          //  Peters 9".2231 

Mean  Obliquity  of  the  ecliptic  for  1750  -f  t, 

according  to  Bessel        ....        23°  28'  18".00  —  0".48368*  —  0".00000272295< 
Mean  Obliquity  of  the  ecliptic  for  1800  +  t, 

according  to  Struve  and  Peters      .        .        23°  27'  54".22  —  0".4738<  —  0".0000014*2 

General  Precession  for  the  year  1750  +  t,  according  to  Bessel          50".21129  +  0".00024429f>6< 
/,  //         u  it  'i         ii  Struve          50".22980  +  0".000226* 


MASSES  OF  THE  PLANETS,  THE  MASS  OP  THE  SUN  BEING  THE  UNIT. 


TO  —  ,       Jupiter 

_        1 

4865751 

~  1047.879' 

1 

1 

VWlllH 

,        Saturn 

390000 

*       *       "       '          3501.6 

1 

1 

TT.artTi 

354936 

*       *       *       '       '      24905 

me 

1 

1 

Mars      .        .        . 

3107713 

*       -           18780 

649 

EXPLANATION  OF  THE  TABLES. 


TABLE  I.  contains  the  values  of  the  angle  of  the  vertical  and  of  the 
logarithm  of  the  earth's  radius,  with  the  geographical  latitude  as  the 
argument.  The  adopted  elements  are  those  derived  by  Bessel.  De- 
noting by  p  the  radius  of  the  earth,  by  <p  the  geographical  latitude, 
and  by  <pr  the  geocentric  latitude,  we  have 

y,  =  <p  —  IV  30".65  sin  2?  +  1".16  sin  4?  —  &c., 

log  p  =  9.9992747  +  0.0007271  cos  2?  —  0.0000018  cos  4?  +  &c., 

f>  being  expressed  in  parts  of  the  equatorial  radius  as  the  unit.  These 
quantities  are  required  in  the  determination  of  the  parallax  of  a 
heavenly  body.  The  formula  for  the  parallax  in  right  ascension  and 
in  declination  are  given  in  Art.  61. 

TABLE  II.  gives  the  intervals  of  sidereal  time  corresponding  to 
given  intervals  of  mean  time.  It  is  required  for  the  conversion  of 
mean  solar  into  sidereal  time. 

TABLE  III.  gives  the  intervals  of  mean  time  corresponding  to 
given  intervals  of  sidereal  time.  It  is  required  for  the  conversion 
t)f  sidereal  into  mean  solar  time. 

TABLE  IV.  furnishes  the  numbers  required  in  converting  hours, 
minutes,  and  seconds  into  decimals  of  a  day.  Thus,  to  convert 
I3h  19m  43.5s  into  the  decimal  of  a  day,  we  find  from  the  Table 

13A    =0.5416667 

19m   =0.0131944 

43s     =  0.0004977 

0.5s  =  0.0000058 

Therefore  ISA  19m  43.5s  =  0.5553646 

861 


652  THEORETICAL   ASTRONOMY. 

The  decimal  corresponding  to  0.5s  is  found  from  that  for  5s  by 
changing  the  place  of  the  decimal  point. 

TABLE  V.  serves  to  find,  for  any  instant,  the  number  of  days  from 
the  beginning  of  the  year.  Thus,  for  1863  Sept.  14,  15h  53m  37.2s, 
we  have 

Sept.  0.0  =  243.00000  days  from  the  beginning  of  the  year. 
Ud  15/i  53m  37.2s  =   14.66224 
Required  number  of  days  =  257.66224 

TABLE  VI.  contains  the  values  of  M  =  75  tan  %v  -f  25  tan3  \v  for 
values  of  v  at  intervals  of  one  minute  from  0°  to  180°.  For  an  ex- 
planation of  its  construction  and  use,  see  Articles  22,  27,  29,  41, 
and  72. 

In  the  case  of  parabolic  motion  the  formulae  are 

m==f  M     mO     T\ 

wherein  log  C0  =  9.9601277.  From  these,  by  means  of  the  Table,  v 
may  be  found  when  t  —  T  is  given,  or  t  —  T  when  v  is  known.  From 
v  =  30°  to  v  =  !SO°  the  Table  contains  the  values  of  log  M. 

TABLE  VII.,  the  construction  of  which  is  explained  in  Art.  23, 
serves  to  determine,  in  the  case  of  parabolic  motion,  the  true  anomaly 
or  the  time  from  the  perihelion  when  v  approaches  near  to  180°. 
The  formulae  are 


8/200 
=Y- 


w  being  taken  in  the  second  quadrant.  The  Table  gives  the  values 
of  AO  with  w  as  the  argument.  As  an  example,  let  it  be  required  to 
find  the  true  anomaly  corresponding  to  the  values  t  —  T=  22.5  days 
and  log  g  =  7.902720.  From  these  we  derive 

log  M  =  4.4582302. 

Table  VI.  gives  for  this  value  of  log  M,  taking  into  account  the 
second  differences, 

i>  =  168°59'32".49; 

but,  using  Table  VII.,  we  have 

w  =  168°  59'  29".ll,  AO  =  3".37, 


EXPLANATION  OF   THE   TABLES.  653 

and  hence 

v  =  w  +  AO  =  168°  59'  32".48, 

the  two  results  agreeing  completely. 

TABLE  VIII.  serves  to  find  the  time  from  the  perihelion  in  the 
case  of  parabolic  motion.  For  an  explanation  of  its  construction 
and  use,  see  Articles  24,  69,  and  72. 

TABLE  IX.  is  used  in  the  determination  of  the  true  anomaly  01 
the  time  from  the  perihelion  in  the  case  of  orbits  of  great  eccen- 
tricity. Its  construction  is  fully  explained  in  Art.  28,  and  its  use  in 
Art.  41. 

TABLE  X.  serves  to  find  the  value  of  v  or  of  t  —  T  in  the  case  of 
elliptic  or  hyperbolic  orbits.  The  construction  of  this  Table  is  ex- 
plained in  Art.  29.  The  first  part  gives  the  values  of  log  B  and 
log  C9  with  A  as  the  argument,  for  the  ellipse  and  the  hyperbola. 
In  the  case  of  log  C  there  are  given  also  log  I.  Diff.  and  log  half  II. 
Diif.,  expressed  in  units  of  the  seventh  decimal  place,  by  means  of 
which  the  interpolation  is  facilitated.  Thus,  if  we  denote  by  log  (C) 
the  value  which  the  Table  gives  directly  for  the  argument  next  less 
than  the  given  value  of  A,  and  by  &A  the  difference  between  this 
argument  and  the  given  value  of  A,  expressed  in  units  of  the  second 
decimal  place,  we  have,  for  the  required  value, 

log  0=  log  (0)  +  Aj.  X  I.  Diff.  +  AJ.2  X  half  II.  Diff. 

For  example,  let  it  be  required  to  find  the  value  of  log  C  correspond- 
ing to  A  =  0.02497 944,  and  the  process  will  be: — 

(1)  (2) 

Arg.  0.02,  log  (C)  =  0.0034986      log  I.  Diff. =4.24585  log  half  II.Diff.  =  1.778 

^  _         8770.6   log*  A     =9.69718  21ogA4  —9.394 

A4  =  0.497944,         (2)= 14.8  3.94303  1.172 

log  0  =  0.0043771 

The  second  part  of  the  Table  gives  the  values  of  A  corresponding 
to  given  values  of  r. 

TABLE  XI.  serves  to  determine  the  chord  of  the  orbit  when  the 
extreme  radii-vectores  and  the  time  of  describing  the  parabolic  arc 
are  given.  For  an  explanation  of  the  construction  and  use  of  this 
Table,  see  Articles  68,  72,  and  117. 


654  THEORETICAL   ASTRONOMY. 

TABLE  XII.  exhibits  the  limits  of  the  real  roots  of  the  equation 

sin  (/  —  C)  =  m0  sin*  /. 

The  construction  and  use  of  this  table  are  fully  explained  in  Articles 
84  and  93. 

TABLES  XIII.  and  XIV.  are  used  in  finding  the  ratio  of  the 
sector  included  by  two  radii-vectores  to  the  triangle  included  by  the 
same  radii-vectores  and  the  chord  joining  their  extremities.  For  an 
explanation  of  the  construction  and  use  of  these  Tables,  see  Articles 
88,  89,  93,  and  101. 

TABLE  XV.  is  used  in  the  determination  of  the  chord  of  the  part 
of  the  orbit  described  in  a  given  time  in  the  case  of  very  eccentric 
elliptic  motion,  and  in  the  determination  of  the  interval  of  time 
whenever  the  chord  is  known.  For  an  explanation  of  its  construc- 
tion and  use,  see  Articles  116,  117,  and  119. 

TABLE  XVI.  is  used  in  finding  the  chord  or  the  interval  of  time 
in  the  case  of  hyperbolic  motion.  See  Articles  118  and  119  for  an 
explanation  of  the  use  of  the  Table,  and  also  the  explanation  of 
Table  X.  for  an  illustration  of  the  use  of  the  columns  headed  log  I. 
Diff.  and  log  half  II.  Diff. 

TABLE  XVII.  is  used  in  the  computation  of  special  perturbations 
when  the  terms  depending  on  the  squares  and  higher  powers  of  the 
masses  are  taken  into  account.  For  an  explanation  of  its  construc- 
tion and  use,  see  Articles  157,  165,  166,  170,  and  171. 

TABLE  XVIII.  contains  the  elements  of  the  orbits  of  the  comets 
which  have  been  observed.  These  elements  are:  T}  the  time  of  peri- 
helion passage  (mean  time  at  Greenwich) ;  TT,  the  longitude  of  the 
perihelion;  &,  the  longitude  of  the  ascending  node;  i,  the  inclina- 
tion of  the  orbit  to  the  plane  of  the  ecliptic;  e,  the  eccentricity  of  the 
orbit;  and  q,  the  perihelion  distance.  The  longitudes  for  Nos.  1,  2, 
12,  16,  91,  92,  115,  127,  138,  155, 156, 159, 160, 162,  171, 173-175 
180,  181,  185,  191,  192,  195-199,  201,  203,  204,  207,  208,  212-215, 
217-219,  221-228,  230,  233,  234,  237-248,  251-258,  261-267, 
269-275,  277-279,  are  in  each  case  measured  from  the  mean  equinox 
of  the  beginning  of  the  year.  In  the  case  of  Nos.  134,  146,  172, 
182,  189,  190,  205,  231,  232,  236,  259,  and  268,  the  longitudes  are 


EXPLANATION  OF  THE  TABLES.  655 

pleasured  from  the  mean  equinox  of  the  beginning  of  the  next  year. 
The  longitudes  for  Nos.  19  and  27  are  measured  from  the  me*\n 
equinox  of  1850.0;  for  No.  186,  from  the  mean  equinox  of  July  3; 
for  No.  187,  from  the  mean  equinox  of  Nov.  9;  for  No.  200,  from 
the  mean  equinox  of  July  1 ;  for  No.  202,  from  the  mean  equinox 
of  Oct.  1 ;  for  No.  206,  from  the  mean  equinox  of  Oct.  7 ;  for  No.  211, 
from  the  mean  equinox  of  1848.0;  for  No.  216,  from  the  mean  equi- 
nox of  Feb.  20 ;  for  No.  229,  from  the  mean  equinox  of  April  1 ;  foi 
No.  250,  from  the  mean  equinox  of  Oct.  1 ;  and  for  No.  276,  from 
the  mean  equinox  of  1865  Oct.  4.0. 

Nos.  1,  2,  11,  12,  20,  23,  29,  41,  53,  80,  and  177  give  the  elements 
for  the  successive  appearances  of  Halley's  comet;  Nos.  104, 116, 126, 
143,  149,  157,  167, 170,  176,  178,  183,  194,  210,  220,  235,  249,  and 
260,  those  for  Encke's  comet,  the  longitudes  being  measured  from  the 
mean  equinox  for  the  instant  of  the  perihelion  passage.  Nos.  92, 
127,  159,  172,  196,  and  222  give  the  elements  for  the  successive  ap- 
pearances of  Biela's  comet;  Nos.  187,  216,  250,  and  276,  those  for 
Faye's  comet;  Nos.  197  and  238,  those  for  Brorsen's  comet;  Nos. 
217  and  243,  those  for  D' Arrest's  comet;  and  Nos.  145  and  245, 
those  for  Winnecke's  comet.  For  epochs  previous  to  1583  the  dates 
are  given  according  to  the  old  style. 

This  Table  is  useful  for  identifying  a  comet  which  may  appear 
with  one  previously  observed,  by  means  of  a  similarity  of  the  ele- 
ments, its  periodic  character  being  otherwise  unknown  or  at  least  un- 
certain. The  elements  given  are  those  which  appear  to  represent  the 
observations  most  completely.  For  a  collection  of  elements  by  vari- 
ous computers,  and  also  for  information  in  regard  to  the  observations 
made  and  in  regard  to  the  place  and  manner  of  their  publication, 
consult  Carl's  Repertorium  der  Cometen-Astronomie  (Munich,  1864), 
or  Galle's  Cometen-Verzeichniss  appended  to  the  latest  edition  of 
Olbers's  Meihode  die  Bahn  eines  Cometen  zu  berechnen. 

TABLE  XIX.  contains  the  elements  of  the  orbits  of  the  minor 
planets,  derived  chiefly  from  the  Berliner  Aslronomisches  Jahrbiivh 
fur  1868.  The  epoch  is  given  in  Berlin  mean  time ;  M  denotes  the 
mean  anomaly,  <p  the  angle  of  eccentricity,  p  the  mean  daily  motion, 
and  a  the  semi-transverse  axis.  The  elements  of  Vesta,  Iris,  Flora, 
Metis,  Victoria,  Eunomia,  Melpomene,  Lutetia,  Proserpina,  and 
Pomona  are  mean  elements;  the  others  are  osculating  for  the  epoch. 
The  date  of  the  discovery  of  the  planet,  and  the  name  of  the  dis- 
coverer, are  also  added. 


THEORETICAL   ASTRONOMY. 

TABLE  XX.  contains  the  mean  elements  of  the  oibits  of  the 
major  planets,  together  with  the  amount  of  their  variations  during  a 
period  of  one  hundred  years.  The  epoch  is  expressed  in  Greenwich 
mean  time,  and  L  denotes  the  mean  longitude  of  the  planet. 

TABLE  XXI.  gives  the  values  of  the  masses  of  the  major  planets, 
and  also  various  constants  which  are  used  in  astronomical  calcula- 
tions. 


APPENDIX. 


A.  Precession.  —  If  we  adopt  the  values  for  the  precession  and  for 
the  variation  of  the  position  of  the  plane  of  the  ecliptic  given  in 
Art.  40,  and  put 

M=  171°  36'  10"  +  39".79  (t  —  1750), 

the  formulae  for  the  annual  precession  in  longitude  (^)  and  latitude 
(/?)  become,  for  the  instant  t, 

~  =  50".2113  +  0".0002443  (t  —  1750) 

-|-  (0".4889  —  0".00000614  (*  —  1750))  cos  (A  —  M)  tan  &        (1) 
^  =  —  (0".4889  —  0".00000614  (*  —  1750))  sin  (A  —  Jf). 

If  we  denote  the  planetary  precession  by  a,  the  luni-solar  preces- 
sion by  ln  and  the  obliquity  of  the  fixed  ecliptic,  at  the  time  1750  -f  rf 
by  e0,  we  have,  according  to  Bessel, 

~  =  0".17926  —  0".0005320786  r, 

^L  =  50^,37572  —  0".000243589  r, 
at 

e0  =  23°  28'  18".0  +  0^.0000098423  T», 

and  if  we  put 

dl,        da  .       dl, 


the  formulae  for  the  annual   precession  in  right  ascension  (a)  and 
declination  (8)  become 


ctt  u  f,y 

—  -  =  m  4-  n  tan  8  sin  a,  -jr  =  n  cos  a,  (  2) 

at  cw 


42 


658  THEORETICAL   ASTRONOMY. 

and  the  numerical  values  of  m  and  n  are,  for  the  instant  £, 

m  =  46".02824  +  0".0003086450  (t  —  1750), 
n  =  20".06442  —  0".0000970zv*  (i  — 


To  determine  the  precession  during  the  interval  t'  —  t,  we  compote 
the  annual  variation  for  the  instant  J  (tf  +  t)  and  this  variation  mul- 
tiplied by  tf  —  t  furnishes  the  required  result. 

jB.  Nutation.  —  The  expressions  for  the  equation  of  the  equinoxes 
and  for  the  nutation  of  the  obliquity  of  the  ecliptic  are,  according  to 
Peters, 

A2  =  —  17".2405  sin  &  -f  0".2073  sin  2&  —  0".2041  sin  2  £  -f  0".0677  sin  (  C  —  F) 

—  l".2694sin  20  +  0".1279  sin  (Q  —  r) 

—  0".0213sin(0  +  r), 

Ae  =  +  9".2231  cos  &  —  0".0897  cos  2&  +  0".0886  cos  2<C 

+  0".5510  cos  20  +  0".0093  cos  (0  +  r), 

for  the  year  1800,  and 

AX  =  —  17".2577  sin  £  +  0".2073  sin  2&  —  0".2041  sin  2  C  -f  0".0677  sin  (  C  —  F  ) 

—  1".2695  sin  2Q  +  0".1275  sin  (0  —  r) 

—  0".0213  sin  (Q  +  r), 

Ae  =  +  9".2240  cos  &  —  0".0896  cos  2^  -f  0".0885  cos  2C 

+  0".5507  cos  20  +  0".0092  cos  (0  +  r), 

for  the  year  1900.  In  these  equations  &  denotes  the  longitude  of 
the  ascending  node  of  the  moon's  orbit,  referred  to  the  mean  equinox, 
<£  the  true  longitude  of  the  moon,  0  the  true  longitude  of  the  sun,  F 
the  true  longitude  of  the  sun's  perigee,  and  P  the  true  longitude  of 
the  moon's  perigee.  The  values  of  these  quantities  may  be  derived 
from  the  solar  and  lunar  tables,  and  thus  the  required  values  of  A^ 
and  AS  may  be  found.  The  equations  give  the  corrections  for  the 
reduction  from  the  mean  equinox  to  the  true  equinox. 

To  find  the  nutation  in  right  ascension  and  in  declination,  if  we 
consider  only  the  terms  of  the  first  order,  we  have 

da         ,    da 

Att=^rAA  +  -^^ 

dS          ,    dd  W 

*$  =  -TT  ^  +  ~J~  Aff- 

cW  ds 

The  values  of  A/I  and  A£  are  found  from  the  preceding  equations,  and 
for  the  differential  coefficients  we  have 


APPENDIX.  659 


do,  dd 

-~jY  =  cos  e  -j-  sin  e  tan  o  sin  a,  -r—  =  cos  a  sin  e, 


-j-  =  —  cos  a  tan  d,  -T-  =  sin  a. 

as  ds 


(6) 


The  terms  of  the  second  order  are  of  sensible  magnitude  only  when 
the  body  is  very  near  the  pole,  and  in  this  case  by  computing  the 
second  differential  coefficients  the  complete  values  may  be  found. 

In  the  reduction  of  the  place  of  a  planet  or  comet  from  the  mean 
equinox  of  one  date  t  to  the  true  equinox  of  another  date  t',  the 
determination  of  the  correction  for  precession  and  of  that  for  nutation 
may  be  effected  simultaneously.  Thus,  let  r  denote  the  interval 
tf  —  t  expressed  in  parts  of  a  year,  and  the  sum  of  the  corrections  for 
precession  and  nutation  gives 

Aa  =  mr  -f-  AA  cos  s  -j-  (nr  -j-  AA  sin  e)  sin  a  tan  d  —  AS  cos  a  tan  «J, 
A<5  =  (nr  -j-  AA  sin  e)  cos  a  -{-AS  sin  a. 

Let  us  now  put 

mr  -f  AA  cos  e  =/, 

nr  -f  AA  sin  s  =  g  cos  G,  (7) 

—  Ae          =  g  sin  G, 

and  the  equations  (6)  become 

*«  =/+  ^sin  (G  +  o)  tan  3, 

A<?=  £COS(£-fa), 

as  already  given  in  Art.  40. 

The  astronomical  ephemerides  give  at  intervals  of  a  few  days  the 
values  of  the  quantities/,  g,  and  G  for  the  reduction  of  the  place  of 
the  body  from  the  mean  equinox  of  the  beginning  of  the  year  to  the 
true  equinox  of  the  date;  and,  in  order  to  obtain  uniformity  and 
accuracy,  the  beginning  of  the  year  is  taken  at  the  instant  when  the 
mean  longitude  of  the  sun  is  280°.  When  these  tables  are  not  avail- 
able, the  values  of  /,  g,  and  G  may  be  found  directly  by  means  of 
the  equations  (7).  The  reduction  from  the  true  equinox  of  t'  to  the 
mean  equinox  of  t  will  be  obtained  by  changing  the  signs  of  the 
corrections. 

C.  Aberration.  —  The  aberration  in  the  case  of  the  planets  and 
comets  may  be  considered  in  three  different  modes  :  — 

1.  If  we  subtract  from  the  observed  time  the  interval  occupied  by 


660  THEORETICAL   ASTRONOMY. 

the  light  in  passing  to  the  earth,  the  result  will  be  the  time  for  which 
the  true  place  is  identical  with  the  apparent  place  for  the  observed 
time. 

2.  If  we  compute   the  time  occupied  by  light  in  traversing  the 
distance  between  the  body  and  the  earth,  and,  by  means  of  the  rate 
of  the  variation  of  the  geocentric  spherical  co-ordinates,  compute  the 
mot.ion  during  this  interval,  we  may  derive  the  true  place  at  the  in- 
stant of  observation. 

3.  We  may  consider  the  observed  place  corrected  for  the  aberration 
of  the  fixed  stars  as  the  true  place  at  the  instant  when  the  light  was 
emitted,  but  as  seen  from  the  place  of  the  earth  at  the  instant  of 
observation. 

The  formula  for  the  actual  aberration  of  the  fixed  stars  are  — 

AA  =  —  20".4451  cos  (A  —  0)  sec  ft  —  0".3429  cos  (A  —  JT)  sec  ft 

*p  =  +  20".4451  sin  (X  —  0  )  sin  ft  +  0".3429  sin  (A  —  /')  sin  f),    W) 

in  the  case  of  the  longitude  and  latitude,  and 

A«  =  —  20".4451  (cos  Q  cos  e  cos  a  -f  sin  0  sin  a)  sec  8 

—  0".3429  (cos  F  cos  e  cos  a  +  sin  ^sin  °0  sec  d, 

Afl  =  -f-  20".4451  cos  Q  (sin  a  sin  d  cos  e  —  cos  <5  sin  e)          (10) 

—  20".4451  sin  0  cos  a  sin  d 

-f-  0".3429  cos  F  (sin  a  sin  d  cos  e  —  cos  d  sin  e) 

—  0".3429  sin  F  cos  a  sin  d, 

in  the  case  of  the  right  ascension  and  declination.  In  these  formulae 
r  denotes  the  longitude  of  the  sun's  perigee,  and  they  give  the  cor- 
rections for  the  reduction  from  the  true  place  to  the  apparent  place. 

D.  Intensity  of  Light.  —  If  we  denote  by  r  the  distance  of  a  planet 
or  comet  from  the  sun,  by  J  its  distance  from  the  earth,  and  by  C  a 
constant  quantity  depending  on  the  magnitude  of  the  body  and  on  its 
capacity  for  reflecting  the  light,  the  intensity  of  the  light  of  the  body 
as  seen  from  the  earth  will  be 

i=--  cm 


When  the  constant  C  is  unknown,  we  may  determine  the  relative 
brilliancy  of  the  comet  at  different  times  by  means  of  the  formula 


APPENDIX.  661 

In  the  case  of  the  planets  we  adopt  as  the  unit  of  the  intensity  of 
light  the  value  of  I  when  the  planet  is  in  opposition  and  both  it  and 
the  earth  are  at  their  mean  distances  from  the  sun.  Thus  we  obtain 

C  =«'(«-!)', 
and  hence 


Let  us  now  denote  by  R  the  ratio  of  the  intensities  of  the  light 
for  two  consecutive  stellar  magnitudes;  then,  if  we  denote  by  M  the 
apparent  stellar  magnitude  of  the  planet  when  J=  1,  and  by  m  the 
magnitude  for  any  value  of  J,  we  shall  have 


R* 

I=SB* 

and  hence 


By  means  of  photometric  determinations  of  the  relative  brilliancy 
of  the  stars,  it  has  been  found  that 

R  =  2.56, 
and  hence  we  derive 

m=M—  2.45  log/,  (15) 

by  means  of  which  the  apparent  stellar  magnitude  of  a  planet  may 
be  determined,  I  being  found  by  means  of  equation  (13).  The  value 
of  M  must  be  determined  for  each  planet  by  means  of  observed  values 
of  m. 

EXAMPLE.  —  The  value  of  M  for  Eurynome  is  10.4;  required  the 
apparent  stellar  magnitude  of  the  planet  when  log  a  =  0.38  795, 
log  r  =  0.2956,  and  log  A  =  9.9952. 

The  equation  (13)  gives 

log  1=  0.5129, 

and  from  (15)  we  derive 

m  =  10.4  —  1.3  =  9.1. 
For  the  values  log  r  =  0.4338,  log  A  =  0.2357,  we  obtain 

log/  =9.7555  —  10, 

and 

m  =  10.4  +  2.45  X  0.2445  =  11.0. 


662  THEORETICAL   ASTRONOMY. 

E.  Numerical  Calculations. — The  extended  numerical  calculations 
required  in  many  of  the  problems  of  Theoretical  Astronomy,  render 
it  important  that  a  judicious  arrangement  of  the  details  should  be 
effected.  The  beginner  will  not,  in  general,  be  able  to  effect  such 
an  arrangement  at  the  outset ;  and  it  would  only  confuse  to  attempt 
to  give  any  specific  directions.  Familiarity  with  the  formulae  to  be 
applied,  and  practice  in  the  performance  of  calculations  of  this 
character,  will  speedily  suggest  those  various  devices  of  arrangement 
by  which  skillful  computers  expedite  the  mechanical  part  of  the 
solution.  There  are,  however,  a  few  general  suggestions  which  may 
be  of  service.  Thus,  it  will  always  facilitate  the  calculation,  when 
several  values  of  a  variable  are  to  be  computed,  to  arrange  it  so  that 
the  values  of  each  function  involved  shall  appear  in  the  same  verti- 
cal or  horizontal  column.  The  course  of  the  differences  will  then 
indicate  the  existence  of  errors  which  might  not  otherwise  be  dis- 
covered until  the  greater  part  if  not  the  entire  calculation  has  been 
completed;  and,  besides,  by  carrying  along  the  several  parts  simulta- 
neously the  use  of  the  logarithmic  and  other  tables  will  be  facilitated. 
Numbers  which  are  to  be  frequently  used  may  be  written  on  slips  of 
paper  and  applied  wherever  they  may  be  required ;  and  by  performing 
the  addition  or  subtraction  of  two  logarithms  or  of  two  numbers  from 
left  to  right  (which  will  be  effected  easily  and  certainly  after  a  little 
practice),  the  sum  or  difference  to  be  used  as  the  argument  in  the 
tables  may  be  retained  in  the  memory,  and  thus  the  required  number 
or  arc  may  be  written  down  directly.  The  number  of  the  decimal 
figures  of  the  logarithms  to  be  used  will  depend  on  the  character  of 
the  data  as  well  as  on  the  accuracy  sought  to  be  obtained,  and  the  use 
of  approximate  formula?  will  be  governed  by  the  same  considerations. 
Whenever  the  formulae  furnish  checks  or  tests  of  the  accuracy  of  the 
numerical  process,  they  should  be  applied;  and  whenever  these  are 
not  provided,  the  use  of  differences  for  the  same  purpose  should  not 
be  overlooked.  By  proper  attention  to  these  suggestions,  much  time 
and  labor  will  be  saved.  The  agreement  of  the  several  proofs  will 
beget  confidence,  relieve  the  mind  from  much  anxiety,  and  thus 
greatly  facilitate  the  progress  of  the  work. 


THE    END. 


GENERAL  LIBRARY 
UNIVERSITY  OF  CALIFORNIA— BERKELEY 

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JUHJ.719MMI 


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JULg    1960 


REC'D 

NOV24196Z 


REC'D  LD 

6Jan's'*p'    F£B4   '65 -10  AM 
DEC  10 1963 

RE^'D  LD* 

Fr35    1964 


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FED    51964 


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LD  21-100m-l,'54(1887sl6)476 


OCT251964 


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